Research ArticleCirculant Type Matrices with the Sum and Product ofFibonacci and Lucas Numbers
Zhaolin Jiang1 Yanpeng Gong2 and Yun Gao2
1 Department of Mathematics Linyi University Linyi Shandong 276000 China2 Institute of Applied Mathematics Shandong University of Technology Zibo Shandong 255049 China
Correspondence should be addressed to Zhaolin Jiang jzh1208sinacom
Received 28 April 2014 Accepted 4 June 2014 Published 19 June 2014
Academic Editor Tongxing Li
Copyright copy 2014 Zhaolin Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Circulant type matrices have become an important tool in solving differential equations In this paper we consider circulanttype matrices including the circulant and left circulant and 119892-circulant matrices with the sum and product of Fibonacci andLucas numbers Firstly we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrixby constructing the transformation matrices Furthermore the invertibility of the left circulant and 119892-circulant matrices is alsodiscussedWe obtain the determinants and the inversematrices of the left circulant and 119892-circulantmatrices by utilizing the relationbetween left circulant and 119892-circulant matrices and circulant matrix respectively
1 Introduction
Circulant matrices may play a crucial role for solving variousdifferential equations In [1] Ruiz-Claeyssen and dos SantosLeal introduced factor circulant matrices matrices with thestructure of circulants butwith the entries below the diagonalbeing multiplied by the same factor The diagonalizationof a circulant matrix and the spectral decomposition areconveniently generalized to block matrices with the structureof factor circulants Matrix and partial differential equationsinvolving factor circulants are considered Wu and Zou in[2] discussed the existence and approximation of solutionsof asymptotic or periodic boundary value problems of mixedfunctional differential equations They focused on (513) in[2] with a circulant matrix whose principal diagonal entriesare zeroes In [3] some Routh-Hurwitz stability conditionsare generalized to the fractional order case The authorsconsidered the 1-system CML (10) They selected a circu-lant matrix which reads a tridiagonal matrix Ahmed andElgazzar used coupled map lattices (CML) as an alternativeapproach to include spatial effects in fractional order systems(FOS) Consider the 1-system CML (10) in [4] They claimedthat the system is stable if all the eigenvalues of the circulantmatrix satisfy (2) in [4] Trench considered nonautonomous
systems of linear differential equations (1) in [5] with someconstraint on the coefficientmatrix119860(119905) One case is that119860(119905)is a variable block circulant matrix Kloeden et al adoptedthe simplest approximation schemes for (1) in [6] with theEuler method which reads (5) in [6] They exploited thatthe covariance matrix of the increments can be embeddedin a circulant matrix The total loops can be done by fastFourier transformation which leads to a total computationalcost of 119874(119898 log119898) = 119874(119899 log 119899) Guo et al concerned ongeneric Dn-Hopf bifurcation to a delayed Hopfield-Cohen-Grossberg model of neural networks (517) in [7] where 119879denoted an interconnection matrix They especially assumed119879 is a symmetric circulant matrix Lin and Yang discretizedthe partial integrodifferential equation (PIDE) in pricingoptions with the preconditioned conjugate gradient (PCG)method which constructed the circulant preconditionersBy using FFT the cost for each linear system is 119874(119899 log 119899)where 119899 is the size of the system in [8] Lee et al investigateda high-order compact (HOC) scheme for the general two-dimensional (2D) linear partial differential equation (11) in[9] with a mixed derivative Meanwhile in order to establishthe 2D combined compact difference (CCD2) scheme theyrewrote (11) in [9] into (21) in [9] Towrite theCCD2 systemin a concise style they employed circulant matrix to obtain
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 375251 12 pageshttpdxdoiorg1011552014375251
2 Abstract and Applied Analysis
the corresponding whole CCD2 linear system (210) in [9]whose entries are circulant block
Circulant type matrices have important applications invarious disciplines including image processing communica-tions signal processing encoding solving Toeplitz matrixproblems and least squares problemsThey have been put onfirm basis with the work of Davis [10] Jiang and Zhou [11]and Gray [12]
In [13] the authors pointed out the processes based on theeigenvalue of circulant type matrices with iid entries Thereare discussions about the convergence in probability and indistribution of the spectral norm of circulant typematrices in[14] The 119892-circulant matrices play an important role in vari-ous applications as well For details please refer to [15 16] andthe references therein Ngondiep et al showed the singularvalues of 119892-circulants in [17] In [18 19] the authors gave thelimiting spectral distributions of left circulant matrices
The Fibonacci and Lucas sequences are defined by thefollowing recurrence relations [20 21] respectively
119865119899+2
= 119865119899+1
+ 119865119899
where 1198650= 0 119865
1= 1
119871119899+2
= 119871119899+1
+ 119871119899
where 1198710= 2 119871
1= 1
(1)
For 119899 ge 0 the first few values of the sequences are givenby the following equation
119899 0 1 2 3 4 5 6 7 8 sdot sdot sdot
1198651198990 1 1 2 3 5 8 13 21 sdot sdot sdot
1198711198992 1 3 4 7 11 18 29 47 sdot sdot sdot
(2)
Let 120572 120573 be the roots of characteristic equation 1199092minus119909minus1 =0 then the Binet formulas of the sequences 119865
119899 and 119871
119899 have
the form
119865119899=120572119899
minus 120573119899
120572 minus 120573 119871
119899= 120572119899
+ 120573119899
120572 =1 + radic5
2 120573 =
1 minus radic5
2
(3)
LetF119899= 119865119899sdot119871119899andL
119899= 119865119899+119871119899 so we can get two new
sequencesF119899andL
119899[22]The two sequences are defined by
the following recurrence relations respectively
F119899+2
= 3F119899+1
+F119899 where F
0= 0 F
1= 1
L119899+2
=L119899+1
+L119899 where L
0= 2 L
1= 2
(4)
For 119899 ge 0 the first few values of the sequences are givenby the following equation
119899 0 1 2 3 4 5 6 7 8 sdot sdot sdot
F1198990 1 3 8 21 55 144 377 987 sdot sdot sdot
L1198992 2 4 6 10 16 26 42 68 sdot sdot sdot
(5)
TheF119899is given by the formulaF
119899= (120572119899
1minus120573119899
1)(1205721minus1205731)
where 1205721 1205731are the roots of 1199092 minus 3119909 + 1 = 0L
119899is given by
the formula L119899= 119865119899+ 119871119899= (120572119899
minus 120573119899
)(120572 minus 120573) + (120572119899
+ 120573119899
)where 120572 120573 are the roots of 1199092 minus 119909 minus 1 = 0
Besides some scholars have given various algorithmsfor the determinants and inverses of nonsingular circulant
matrices [10 11] Unfortunately the computational com-plexities of these algorithms are very amazing with theorder of matrix increasing However some authors gave theexplicit determinants and inverse of circulant and skew-circulant involving Fibonacci and Lucas numbers For exam-ple Dazheng gave the determinant of the Fibonacci-Lucasquasicyclic matrices in [20] Shen et al considered circulantmatrices with Fibonacci and Lucas numbers and presentedtheir explicit determinants and inverses by constructingthe transformation matrices [21] Jaiswal evaluated somedeterminants of circulant whose elements are the generalizedFibonacci numbers [23] Lind presented the determinants ofcirculant and skew-circulant involving Fibonacci numbers[24] Bozkurt and Tam gave determinants and inversesof circulant matrices with Jacobsthal and Jacobsthal-Lucasnumbers [25]
In [22] the authors gave some determinantal and perma-nental representations of F
119899and L
119899and complex factor-
ization formulas The purpose of this paper is to obtain theexplicit determinants and inverse of circulant type matricesby some perfect properties ofF
119899andL
119899
In this paper we adopt the following two conventions00
= 1 and for any sequence 119886119899 sum119899119896=119894119886119896= 0 in the case
119894 gt 119899
Definition 1 (see [10 11]) In a circulant matrix (or rightcirculant matrix [26])
each row is a cyclic shift of the row above to the left
Left circulant matrix is a special Hankel matrix
Definition 3 (see [14 27]) A 119892-circulant matrix is an 119899 times 119899complex matrix with the following form
119860119892119899=(
1198861
1198862
sdot sdot sdot 119886119899
119886119899minus119892+1
119886119899minus119892+2
sdot sdot sdot 119886119899minus119892
119886119899minus2119892+1
119886119899minus2119892+2
sdot sdot sdot 119886119899minus2119892
d
119886119892+1
119886119892+2
sdot sdot sdot 119886119892
) (8)
Abstract and Applied Analysis 3
where 119892 is a nonnegative integer and each of the subscripts isunderstood to be reduced modulo 119899
The first row of 119860119892119899
is (1198861 1198862 119886
119899) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by 119892positions (equivalently 119892 mod 119899 positions) Note that 119892 = 1or 119892 = 119899 + 1 yields the standard circulant matrix If 119892 = 119899 minus 1then we obtain the left circulant matrix
Lemma4 (see [21]) Let119860 = Circ(1198861 1198862 119886
119899) be a circulant
matrix then one has
(i) 119860 is invertible if and only if 119891(120596119896) = 0 (119896 = 0 1
2 119899 minus 1) where 119891(119909) = sum119899
119895=1119886119895119909119895minus1 and 120596 =
exp(2120587119894119899)
(ii) If119860 is invertible then the inverse119860minus1 of119860 is a circulantmatrix
Lemma 5 Define
Δ =((
(
1 0 0 sdot sdot sdot 0 0
0 0 0 sdot sdot sdot 0 1
0 0 0 sdot sdot sdot 1 0
d
0 0 1 sdot sdot sdot 0 0
0 1 0 sdot sdot sdot 0 0
))
)
(9)
the matrix Δ is an orthogonal cyclic shift matrix (and aleft circulant matrix) It holds that LCirc(119886
1 1198862 119886
119899) =
ΔCirc(1198861 1198862 119886
119899)
Lemma6 (see [27]) The 119899times119899matrixQ119892is unitary if and only
if (119899 119892) = 1 whereQ119892is a 119892-circulant matrix with the first row
119890lowast
= [1 0 0]
Lemma 7 (see [27]) 119860119892119899
is a 119892-circulant matrix with the firstrow [119886
1 1198862 119886
119899] if and only if 119860
119892119899= Q119892119862 where 119862 =
Circ(1198861 1198862 119886
119899)
2 Determinant and Inverse of a CirculantMatrix with the Product of the Fibonacciand Lucas Numbers
In this section letA119899= Circ(F
1F2 F
119899) be a circulant
matrix Firstly we give the determinant equation of matrixA119899 Afterwards we prove that A
119899is an invertible matrix for
119899 gt 2 and then we find the inverse of the matrixA119899
Theorem 8 Let A119899= Circ(F
1F2 F
119899) be a circulant
matrix then one has
detA119899= (1 minusF
119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(10)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Proof Obviously detA1= 1 satisfies (10) In the case 119899 gt 1
let
Γ =
((((((
(
1
minus3 1
1 1 minus3
0 0 1 minus3 1
c c c0 1 c c0 1 minus3 c 0
0 1 minus3 1
))))))
)119899times119899
Π1=
(((((((((
(
1 0 0 sdot sdot sdot 0 0
0 (minusF119899
F1minusF119899+1
)
119899minus2
0 sdot sdot sdot 0 1
0 (minusF119899
F1minusF119899+1
)
119899minus3
0 sdot sdot sdot 1 0
d
0minusF119899
F1minusF119899+1
1 sdot sdot sdot 0 0
0 1 0 sdot sdot sdot 0 0
)))))))))
)119899times119899
(11)
We can obtain
ΓA119899Π1
=((
(
F11198911015840
119899F119899minus1
F119899minus2
sdot sdot sdot F2
0 119891119899
minusF119899minus2
minusF119899minus3
sdot sdot sdot minusF1
0 0 F1minusF119899+1
0 0 F119899
F1minusF119899+1
F
119899
0 0 0 F1minusF119899+1
))
)
(12)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
1198911015840
119899=
119899minus1
sum
119896=1
F119896+1(
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(13)
We obtain
det Γ detA119899detΠ1
= F1[F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
]
times (F1minusF119899+1)119899minus2
= F1[F1minusF119899+1
+
119899minus1
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
]
4 Abstract and Applied Analysis
times (F1minusF119899+1)119899minus2
= (1 minusF119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(14)
while
det Γ = detΠ1= (minus1)
(119899minus1)(119899minus2)2
(15)
we have
detA119899= (1 minusF
119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(16)
Thus the proof is completed
Theorem 9 Let A119899= Circ(F
1F2 F
119899) be a circulant
matrix if 119899 gt 2 thenA119899is an invertible matrix
Proof When 119899 = 3 in Theorem 8 we have detA3= 468 = 0
henceA3is invertible In the case 119899 gt 3 sinceF
119899= (120572119899
1minus120573119899
1)
(1205721minus 1205731) where 120572
1+ 1205731= 3 120572
1sdot 1205731= 1We have
119891 (120596119896
) =
119899
sum
119895=1
F119895(120596119896
)119895minus1
=1
1205721minus 1205731
119899
sum
119895=1
(120572119895
1minus 120573119895
1) (120596119896
)119895minus1
=1
1205721minus 1205731
[1205721(1 minus 120572
119899
1)
1 minus 1205721120596119896
minus1205731(1 minus 120573
119899
1)
1 minus 1205731120596119896
]
=1
1205721minus 1205731
[(1205721minus 1205731) minus (120572
119899+1
1minus 120573119899+1
1)
1 minus (1205721+ 1205731) 120596119896 + 120572
112057311205962119896
]
+1
1205721minus 1205731
[12057211205731(120572119899
1minus 120573119899
1) 120596119896
1 minus (1205721+ 1205731) 120596119896 + 120572
112057311205962119896
]
=1 minusF
119899+1+F119899120596119896
1 minus 3120596119896 + 1205962119896(119896 = 1 2 119899 minus 1)
(17)
If there exists 120596119897 (119897 = 1 2 119899 minus 1) such that 119891(120596119897) = 0we obtain 1minusF
119899+1+F119899120596119897
= 0 for 1minus3120596119897+1205962119897 = 0 thus 120596119897 =(F119899+1
minus 1)F119899is a real number While 120596119897 = exp(2119897120587119894119899) =
cos(2119897120587119899) + 119894 sin(2119897120587119899) hence sin(2119897120587119899) = 0 so we have120596119897
= minus1 for 0 lt 2119897120587119899 lt 2120587 But 119909 = minus1 is not the root ofequation 1 minusF
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
the corresponding whole CCD2 linear system (210) in [9]whose entries are circulant block
Circulant type matrices have important applications invarious disciplines including image processing communica-tions signal processing encoding solving Toeplitz matrixproblems and least squares problemsThey have been put onfirm basis with the work of Davis [10] Jiang and Zhou [11]and Gray [12]
In [13] the authors pointed out the processes based on theeigenvalue of circulant type matrices with iid entries Thereare discussions about the convergence in probability and indistribution of the spectral norm of circulant typematrices in[14] The 119892-circulant matrices play an important role in vari-ous applications as well For details please refer to [15 16] andthe references therein Ngondiep et al showed the singularvalues of 119892-circulants in [17] In [18 19] the authors gave thelimiting spectral distributions of left circulant matrices
The Fibonacci and Lucas sequences are defined by thefollowing recurrence relations [20 21] respectively
119865119899+2
= 119865119899+1
+ 119865119899
where 1198650= 0 119865
1= 1
119871119899+2
= 119871119899+1
+ 119871119899
where 1198710= 2 119871
1= 1
(1)
For 119899 ge 0 the first few values of the sequences are givenby the following equation
119899 0 1 2 3 4 5 6 7 8 sdot sdot sdot
1198651198990 1 1 2 3 5 8 13 21 sdot sdot sdot
1198711198992 1 3 4 7 11 18 29 47 sdot sdot sdot
(2)
Let 120572 120573 be the roots of characteristic equation 1199092minus119909minus1 =0 then the Binet formulas of the sequences 119865
119899 and 119871
119899 have
the form
119865119899=120572119899
minus 120573119899
120572 minus 120573 119871
119899= 120572119899
+ 120573119899
120572 =1 + radic5
2 120573 =
1 minus radic5
2
(3)
LetF119899= 119865119899sdot119871119899andL
119899= 119865119899+119871119899 so we can get two new
sequencesF119899andL
119899[22]The two sequences are defined by
the following recurrence relations respectively
F119899+2
= 3F119899+1
+F119899 where F
0= 0 F
1= 1
L119899+2
=L119899+1
+L119899 where L
0= 2 L
1= 2
(4)
For 119899 ge 0 the first few values of the sequences are givenby the following equation
119899 0 1 2 3 4 5 6 7 8 sdot sdot sdot
F1198990 1 3 8 21 55 144 377 987 sdot sdot sdot
L1198992 2 4 6 10 16 26 42 68 sdot sdot sdot
(5)
TheF119899is given by the formulaF
119899= (120572119899
1minus120573119899
1)(1205721minus1205731)
where 1205721 1205731are the roots of 1199092 minus 3119909 + 1 = 0L
119899is given by
the formula L119899= 119865119899+ 119871119899= (120572119899
minus 120573119899
)(120572 minus 120573) + (120572119899
+ 120573119899
)where 120572 120573 are the roots of 1199092 minus 119909 minus 1 = 0
Besides some scholars have given various algorithmsfor the determinants and inverses of nonsingular circulant
matrices [10 11] Unfortunately the computational com-plexities of these algorithms are very amazing with theorder of matrix increasing However some authors gave theexplicit determinants and inverse of circulant and skew-circulant involving Fibonacci and Lucas numbers For exam-ple Dazheng gave the determinant of the Fibonacci-Lucasquasicyclic matrices in [20] Shen et al considered circulantmatrices with Fibonacci and Lucas numbers and presentedtheir explicit determinants and inverses by constructingthe transformation matrices [21] Jaiswal evaluated somedeterminants of circulant whose elements are the generalizedFibonacci numbers [23] Lind presented the determinants ofcirculant and skew-circulant involving Fibonacci numbers[24] Bozkurt and Tam gave determinants and inversesof circulant matrices with Jacobsthal and Jacobsthal-Lucasnumbers [25]
In [22] the authors gave some determinantal and perma-nental representations of F
119899and L
119899and complex factor-
ization formulas The purpose of this paper is to obtain theexplicit determinants and inverse of circulant type matricesby some perfect properties ofF
119899andL
119899
In this paper we adopt the following two conventions00
= 1 and for any sequence 119886119899 sum119899119896=119894119886119896= 0 in the case
119894 gt 119899
Definition 1 (see [10 11]) In a circulant matrix (or rightcirculant matrix [26])
each row is a cyclic shift of the row above to the left
Left circulant matrix is a special Hankel matrix
Definition 3 (see [14 27]) A 119892-circulant matrix is an 119899 times 119899complex matrix with the following form
119860119892119899=(
1198861
1198862
sdot sdot sdot 119886119899
119886119899minus119892+1
119886119899minus119892+2
sdot sdot sdot 119886119899minus119892
119886119899minus2119892+1
119886119899minus2119892+2
sdot sdot sdot 119886119899minus2119892
d
119886119892+1
119886119892+2
sdot sdot sdot 119886119892
) (8)
Abstract and Applied Analysis 3
where 119892 is a nonnegative integer and each of the subscripts isunderstood to be reduced modulo 119899
The first row of 119860119892119899
is (1198861 1198862 119886
119899) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by 119892positions (equivalently 119892 mod 119899 positions) Note that 119892 = 1or 119892 = 119899 + 1 yields the standard circulant matrix If 119892 = 119899 minus 1then we obtain the left circulant matrix
Lemma4 (see [21]) Let119860 = Circ(1198861 1198862 119886
119899) be a circulant
matrix then one has
(i) 119860 is invertible if and only if 119891(120596119896) = 0 (119896 = 0 1
2 119899 minus 1) where 119891(119909) = sum119899
119895=1119886119895119909119895minus1 and 120596 =
exp(2120587119894119899)
(ii) If119860 is invertible then the inverse119860minus1 of119860 is a circulantmatrix
Lemma 5 Define
Δ =((
(
1 0 0 sdot sdot sdot 0 0
0 0 0 sdot sdot sdot 0 1
0 0 0 sdot sdot sdot 1 0
d
0 0 1 sdot sdot sdot 0 0
0 1 0 sdot sdot sdot 0 0
))
)
(9)
the matrix Δ is an orthogonal cyclic shift matrix (and aleft circulant matrix) It holds that LCirc(119886
1 1198862 119886
119899) =
ΔCirc(1198861 1198862 119886
119899)
Lemma6 (see [27]) The 119899times119899matrixQ119892is unitary if and only
if (119899 119892) = 1 whereQ119892is a 119892-circulant matrix with the first row
119890lowast
= [1 0 0]
Lemma 7 (see [27]) 119860119892119899
is a 119892-circulant matrix with the firstrow [119886
1 1198862 119886
119899] if and only if 119860
119892119899= Q119892119862 where 119862 =
Circ(1198861 1198862 119886
119899)
2 Determinant and Inverse of a CirculantMatrix with the Product of the Fibonacciand Lucas Numbers
In this section letA119899= Circ(F
1F2 F
119899) be a circulant
matrix Firstly we give the determinant equation of matrixA119899 Afterwards we prove that A
119899is an invertible matrix for
119899 gt 2 and then we find the inverse of the matrixA119899
Theorem 8 Let A119899= Circ(F
1F2 F
119899) be a circulant
matrix then one has
detA119899= (1 minusF
119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(10)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Proof Obviously detA1= 1 satisfies (10) In the case 119899 gt 1
let
Γ =
((((((
(
1
minus3 1
1 1 minus3
0 0 1 minus3 1
c c c0 1 c c0 1 minus3 c 0
0 1 minus3 1
))))))
)119899times119899
Π1=
(((((((((
(
1 0 0 sdot sdot sdot 0 0
0 (minusF119899
F1minusF119899+1
)
119899minus2
0 sdot sdot sdot 0 1
0 (minusF119899
F1minusF119899+1
)
119899minus3
0 sdot sdot sdot 1 0
d
0minusF119899
F1minusF119899+1
1 sdot sdot sdot 0 0
0 1 0 sdot sdot sdot 0 0
)))))))))
)119899times119899
(11)
We can obtain
ΓA119899Π1
=((
(
F11198911015840
119899F119899minus1
F119899minus2
sdot sdot sdot F2
0 119891119899
minusF119899minus2
minusF119899minus3
sdot sdot sdot minusF1
0 0 F1minusF119899+1
0 0 F119899
F1minusF119899+1
F
119899
0 0 0 F1minusF119899+1
))
)
(12)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
1198911015840
119899=
119899minus1
sum
119896=1
F119896+1(
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(13)
We obtain
det Γ detA119899detΠ1
= F1[F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
]
times (F1minusF119899+1)119899minus2
= F1[F1minusF119899+1
+
119899minus1
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
]
4 Abstract and Applied Analysis
times (F1minusF119899+1)119899minus2
= (1 minusF119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(14)
while
det Γ = detΠ1= (minus1)
(119899minus1)(119899minus2)2
(15)
we have
detA119899= (1 minusF
119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(16)
Thus the proof is completed
Theorem 9 Let A119899= Circ(F
1F2 F
119899) be a circulant
matrix if 119899 gt 2 thenA119899is an invertible matrix
Proof When 119899 = 3 in Theorem 8 we have detA3= 468 = 0
henceA3is invertible In the case 119899 gt 3 sinceF
119899= (120572119899
1minus120573119899
1)
(1205721minus 1205731) where 120572
1+ 1205731= 3 120572
1sdot 1205731= 1We have
119891 (120596119896
) =
119899
sum
119895=1
F119895(120596119896
)119895minus1
=1
1205721minus 1205731
119899
sum
119895=1
(120572119895
1minus 120573119895
1) (120596119896
)119895minus1
=1
1205721minus 1205731
[1205721(1 minus 120572
119899
1)
1 minus 1205721120596119896
minus1205731(1 minus 120573
119899
1)
1 minus 1205731120596119896
]
=1
1205721minus 1205731
[(1205721minus 1205731) minus (120572
119899+1
1minus 120573119899+1
1)
1 minus (1205721+ 1205731) 120596119896 + 120572
112057311205962119896
]
+1
1205721minus 1205731
[12057211205731(120572119899
1minus 120573119899
1) 120596119896
1 minus (1205721+ 1205731) 120596119896 + 120572
112057311205962119896
]
=1 minusF
119899+1+F119899120596119896
1 minus 3120596119896 + 1205962119896(119896 = 1 2 119899 minus 1)
(17)
If there exists 120596119897 (119897 = 1 2 119899 minus 1) such that 119891(120596119897) = 0we obtain 1minusF
119899+1+F119899120596119897
= 0 for 1minus3120596119897+1205962119897 = 0 thus 120596119897 =(F119899+1
minus 1)F119899is a real number While 120596119897 = exp(2119897120587119894119899) =
cos(2119897120587119899) + 119894 sin(2119897120587119899) hence sin(2119897120587119899) = 0 so we have120596119897
= minus1 for 0 lt 2119897120587119899 lt 2120587 But 119909 = minus1 is not the root ofequation 1 minusF
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
where 119892 is a nonnegative integer and each of the subscripts isunderstood to be reduced modulo 119899
The first row of 119860119892119899
is (1198861 1198862 119886
119899) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by 119892positions (equivalently 119892 mod 119899 positions) Note that 119892 = 1or 119892 = 119899 + 1 yields the standard circulant matrix If 119892 = 119899 minus 1then we obtain the left circulant matrix
Lemma4 (see [21]) Let119860 = Circ(1198861 1198862 119886
119899) be a circulant
matrix then one has
(i) 119860 is invertible if and only if 119891(120596119896) = 0 (119896 = 0 1
2 119899 minus 1) where 119891(119909) = sum119899
119895=1119886119895119909119895minus1 and 120596 =
exp(2120587119894119899)
(ii) If119860 is invertible then the inverse119860minus1 of119860 is a circulantmatrix
Lemma 5 Define
Δ =((
(
1 0 0 sdot sdot sdot 0 0
0 0 0 sdot sdot sdot 0 1
0 0 0 sdot sdot sdot 1 0
d
0 0 1 sdot sdot sdot 0 0
0 1 0 sdot sdot sdot 0 0
))
)
(9)
the matrix Δ is an orthogonal cyclic shift matrix (and aleft circulant matrix) It holds that LCirc(119886
1 1198862 119886
119899) =
ΔCirc(1198861 1198862 119886
119899)
Lemma6 (see [27]) The 119899times119899matrixQ119892is unitary if and only
if (119899 119892) = 1 whereQ119892is a 119892-circulant matrix with the first row
119890lowast
= [1 0 0]
Lemma 7 (see [27]) 119860119892119899
is a 119892-circulant matrix with the firstrow [119886
1 1198862 119886
119899] if and only if 119860
119892119899= Q119892119862 where 119862 =
Circ(1198861 1198862 119886
119899)
2 Determinant and Inverse of a CirculantMatrix with the Product of the Fibonacciand Lucas Numbers
In this section letA119899= Circ(F
1F2 F
119899) be a circulant
matrix Firstly we give the determinant equation of matrixA119899 Afterwards we prove that A
119899is an invertible matrix for
119899 gt 2 and then we find the inverse of the matrixA119899
Theorem 8 Let A119899= Circ(F
1F2 F
119899) be a circulant
matrix then one has
detA119899= (1 minusF
119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(10)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Proof Obviously detA1= 1 satisfies (10) In the case 119899 gt 1
let
Γ =
((((((
(
1
minus3 1
1 1 minus3
0 0 1 minus3 1
c c c0 1 c c0 1 minus3 c 0
0 1 minus3 1
))))))
)119899times119899
Π1=
(((((((((
(
1 0 0 sdot sdot sdot 0 0
0 (minusF119899
F1minusF119899+1
)
119899minus2
0 sdot sdot sdot 0 1
0 (minusF119899
F1minusF119899+1
)
119899minus3
0 sdot sdot sdot 1 0
d
0minusF119899
F1minusF119899+1
1 sdot sdot sdot 0 0
0 1 0 sdot sdot sdot 0 0
)))))))))
)119899times119899
(11)
We can obtain
ΓA119899Π1
=((
(
F11198911015840
119899F119899minus1
F119899minus2
sdot sdot sdot F2
0 119891119899
minusF119899minus2
minusF119899minus3
sdot sdot sdot minusF1
0 0 F1minusF119899+1
0 0 F119899
F1minusF119899+1
F
119899
0 0 0 F1minusF119899+1
))
)
(12)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
1198911015840
119899=
119899minus1
sum
119896=1
F119896+1(
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(13)
We obtain
det Γ detA119899detΠ1
= F1[F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
]
times (F1minusF119899+1)119899minus2
= F1[F1minusF119899+1
+
119899minus1
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
]
4 Abstract and Applied Analysis
times (F1minusF119899+1)119899minus2
= (1 minusF119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(14)
while
det Γ = detΠ1= (minus1)
(119899minus1)(119899minus2)2
(15)
we have
detA119899= (1 minusF
119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 minusF119899+1
minusF119899
)
119896minus1
(16)
Thus the proof is completed
Theorem 9 Let A119899= Circ(F
1F2 F
119899) be a circulant
matrix if 119899 gt 2 thenA119899is an invertible matrix
Proof When 119899 = 3 in Theorem 8 we have detA3= 468 = 0
henceA3is invertible In the case 119899 gt 3 sinceF
119899= (120572119899
1minus120573119899
1)
(1205721minus 1205731) where 120572
1+ 1205731= 3 120572
1sdot 1205731= 1We have
119891 (120596119896
) =
119899
sum
119895=1
F119895(120596119896
)119895minus1
=1
1205721minus 1205731
119899
sum
119895=1
(120572119895
1minus 120573119895
1) (120596119896
)119895minus1
=1
1205721minus 1205731
[1205721(1 minus 120572
119899
1)
1 minus 1205721120596119896
minus1205731(1 minus 120573
119899
1)
1 minus 1205731120596119896
]
=1
1205721minus 1205731
[(1205721minus 1205731) minus (120572
119899+1
1minus 120573119899+1
1)
1 minus (1205721+ 1205731) 120596119896 + 120572
112057311205962119896
]
+1
1205721minus 1205731
[12057211205731(120572119899
1minus 120573119899
1) 120596119896
1 minus (1205721+ 1205731) 120596119896 + 120572
112057311205962119896
]
=1 minusF
119899+1+F119899120596119896
1 minus 3120596119896 + 1205962119896(119896 = 1 2 119899 minus 1)
(17)
If there exists 120596119897 (119897 = 1 2 119899 minus 1) such that 119891(120596119897) = 0we obtain 1minusF
119899+1+F119899120596119897
= 0 for 1minus3120596119897+1205962119897 = 0 thus 120596119897 =(F119899+1
minus 1)F119899is a real number While 120596119897 = exp(2119897120587119894119899) =
cos(2119897120587119899) + 119894 sin(2119897120587119899) hence sin(2119897120587119899) = 0 so we have120596119897
= minus1 for 0 lt 2119897120587119899 lt 2120587 But 119909 = minus1 is not the root ofequation 1 minusF
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
4 Determinant and Inverse of a Left CirculantMatrix with F
119899and L
119899Numbers
In this section let A1015840119899= LCirc(F
1F2 F
119899) and B1015840
119899=
LCirc(L1L2 L
119899) be left circulant matrices By using the
obtained conclusions we give a determinant formula forthe matrix A1015840
119899and B1015840
119899 Afterwards we prove that A1015840
119899is an
invertible matrix for 119899 gt 2 andB1015840119899is an invertible matrix for
any positive integer 119899The inverses of thematricesA1015840119899andB1015840
119899
are also presentedAccording to Lemma 5 andTheorems 8 9 and 11 we can
obtain the following theorems
Theorem 16 Let A1015840119899= LCirc(F
1F2 F
119899) be a left
circulant matrix then one has
detA1015840119899= (minus1)
(119899minus1)(119899minus2)2
times [(1 +F119899+1)119899minus1
+ (minusF119899)119899minus2
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(54)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 17 LetA1015840119899= LCirc(F
1F2 F
119899) be a left circu-
lant matrix if 119899 gt 2 thenA1015840119899is an invertible matrix
Theorem 18 Let A1015840119899= LCirc(F
1F2 F
119899) (119899 gt 2) be a
left circulant matrix then one has
A1015840minus1
119899=1
119891119899
LCirc(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
(minusF119899)2
(F1minusF119899+1)3
10 Abstract and Applied Analysis
minusF119899
(F1minusF119899+1)2
1
F1minusF119899+1
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894)
(55)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(56)
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
By Lemma 5 and Theorems 12 13 and 15 the followingconclusions can be attained
Theorem 19 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one has
detB1015840119899= 2(minus1)
(119899minus1)(119899minus2)2
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(57)
whereL119899is the 119899th 119865
119899+ 119871119899number
Theorem20 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix thenB1015840119899is invertible for any positive integer 119899
Theorem 21 LetB1015840119899= LCirc(L
1L2 L
119899) be a left circu-
lant matrix then one can obtain
B1015840minus1119899
=1
119897119899
LCirc(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
2(L119899minus 2)2
(L1minusL119899+1)3
2 (L119899minus 2)
(L1minusL119899+1)2
2
L1minusL119899+1
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
)
(58)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(59)
5 Determinant and Inverse of 119892-CirculantMatrix with F
119899and L
119899Numbers
In this section let A119892119899
= 119892-Circ(F1F2 F
119899) and
B119892119899
= 119892-Circ(L1L2 L
119899) be 119892-circulant matrices
By using the obtained conclusions we give a determinantformula for thematricesA
119892119899andB
119892119899 Afterwards we prove
that A119892119899
is an invertible matrix for 119899 gt 2 and B119892119899
is aninvertible matrix if (119899 119892) = 1 The inverse of the matricesA119892119899
andB119892119899
are also presentedFrom Lemmas 6 and 7 and Theorems 8 9 and 11 we
deduce the following results
Theorem 22 LetA119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culant matrix then one has
detA119892119899= detQ
119892[(1 +F
119899+1)119899minus1
+ (minusF119899)119899minus2
times
119899minus1
sum
119896=1
(minusF119896) (
1 +F119899+1
minusF119899
)
119896minus1
]
(60)
whereF119899is the 119899th 119865
119899sdot 119871119899number
Theorem 23 Let A119892119899
= 119892-Circ(F1F2 F
119899) be a 119892-cir-
culantmatrix and (119892 119899) = 1 if 119899 gt 2 thenA119892119899
is an invertiblematrix
Theorem 24 LetA119892119899
= 119892-Circ(F1F2 F
119899) (119899 gt 2) be
a 119892-circulant matrix and (119892 119899) = 1 then
Aminus1
119892119899
= [1
119891119899
Circ(1 minus119899minus2
sum
119894=1
F119899minus119894(minusF119899)119894minus1
(F1minusF119899+1)119894
minus 3 +
119899minus2
sum
119894=1
F119899minus1minus119894
(minusF119899)119894minus1
(F1minusF119899+1)119894
1
F1minusF119899+1
minusF119899
(F1minusF119899+1)2
(minusF119899)2
(F1minusF119899+1)3
(minusF119899)119899minus3
(F1minusF119899+1)119899minus2
)]Q119879
119892
(61)
where
119891119899= F1minus 3F119899+
119899minus2
sum
119896=1
(minusF119896) (
minusF119899
F1minusF119899+1
)
119899minus(119896+1)
(62)
Taking Lemmas 6 and 7 andTheorems 12 13 and 15 intoaccount one has the following theorems
Abstract and Applied Analysis 11
Theorem 25 Let B119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-
circulant matrix then one has
detB119892119899= 2 detQ
119892
times [(2 minusL119899+1)119899minus1
+ (L119899minus 2)119899minus2
times
119899minus1
sum
119896=1
(L119896+2
minus 2L119896+1) (
2 minusL119899+1
L119899minus 2
)
119896minus1
]
(63)
whereL119899is the 119899 119865
119899+ 119871119899number
Theorem 26 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
culant matrix and (119892 119899) = 1 then B119892119899
is invertible for anypositive integer 119899
Theorem 27 LetB119892119899
= 119892-Circ(L1L2 L
119899) be a 119892-cir-
culant matrix and (119892 119899) = 1 then
Bminus1
119892119899
= [1
119897119899
Circ(1 minus119899minus2
sum
119894=1
L119899minus119894minus1
(L119899minus 2)119894minus1
(L1minusL119899+1)119894
minus 2 minus
119899minus2
sum
119894=1
(2L119899minus119894minusL119899minus119894+1
) (L119899minus 2)119894minus1
(L1minusL119899+1)119894
2
L1minusL119899+1
2 (L119899minus 2)
(L1minusL119899+1)2
2(L119899minus 2)2
(L1minusL119899+1)3
2(L119899minus 2)119899minus3
(L1minusL119899+1)119899minus2
)]Q119879
119892
(64)
where
119897119899= L1minus 2L119899
+
119899minus2
sum
119896=1
(L119896+2
minus 2L119896+1) (
L119899minus 2
L1minusL119899+1
)
119899minus(119896+1)
(65)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research was supported by the Development Projectof Science amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] J C Ruiz-Claeyssen and L A dos Santos Leal ldquoDiagonalizationand spectral decomposition of factor block circulant matricesrdquoLinear Algebra and its Applications vol 99 pp 41ndash61 1988
[2] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[3] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006
[4] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 pp 607ndash614 2007
[5] W F Trench ldquoOn nonautonomous linear systems of differentialand difference equations with 119877-symmetric coefficient matri-cesrdquo Linear Algebra and Its Applications vol 431 no 11 pp2109ndash2117 2009
[6] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[7] S J Guo Y M Chen and J H Wu ldquoEquivariant normal formsfor parameterized delay differential equations with applicationsto bifurcation theoryrdquo Acta Mathematica Sinica vol 28 no 4pp 825ndash856 2012
[8] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo A Quarterly on Numerical Analysis and Theory ofComputation vol 50 no 4 pp 313ndash327 2013
[9] S T Lee J Liu and H-W Sun ldquoCombined compact differencescheme for linear second-order partial differential equationswith mixed derivativerdquo Journal of Computational and AppliedMathematics vol 264 pp 23ndash37 2014
[10] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[11] Z L Jiang and Z X Zhou Circulant Matrices Chengdu Tech-nology University Publishing Company Chengdu China 1999
[12] R M Gray ldquoToeplitz and circulant matrices a reviewrdquo Foun-dations and Trends in Communication and Information Theoryvol 2 pp 155ndash239 2006
[13] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes StatisticalTheory and Applications in Science Engineering and Economicsvol 14 no 4 pp 365ndash392 2011
[14] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[15] C Erbas and M M Tanik ldquoGenerating solutions to the 119873-queens problem using 119892-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[16] Y-K Wu R-Z Jia and Q Li ldquo(0 1)-circulant solutions to the119860119898
= 119869119899matrix equation 119892rdquo Linear Algebra and Its Applications
vol 345 pp 195ndash224 2002[17] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectral
features and asymptotic properties for (0 1)-circulants and
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
[18] A Basak and A Bose ldquoLimiting spectral distributions of someband matricesrdquo Periodica Mathematica Hungarica vol 63 no1 pp 113ndash150 2011
[19] A Bose and J Mitra ldquoLimiting spectral distribution of a specialcirculantrdquo Statistics amp Probability Letters vol 60 no 1 pp 111ndash120 2002
[20] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquoThe Fibo-nacci Quarterly vol 40 no 3 pp 280ndash286 2002
[21] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[22] F L Lu and Z L Jiang ldquoThe sum and product of Fibonacci andLucas numbers Pell and Pell-Lucas numbers representation bymatrix methodrdquo WSEAS Transactions on Mathematics vol 12no 4 p 449
[24] D A Lind ldquoA Fibonacci circulantrdquoTheFibonacci Quarterly vol8 no 5 pp 449ndash455 1970
[25] D Bozkurt and T-Y Tam ldquoDeterminants and inverses of circu-lant matrices with Jacobsthal and Jacobsthal-Lucas NumbersrdquoApplied Mathematics and Computation vol 219 no 2 pp 544ndash551 2012
[26] H Karner J Schneid and C W Ueberhuber ldquoSpectral decom-position of real circulant matricesrdquo Linear Algebra and ItsApplications vol 367 pp 301ndash311 2003
[27] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 pp 385ndash388 1972
![CIRCULANT TENSORS WITH APPLICATIONS TO SPECTRAL … · resonance imaging [3,10,25,26] and spectral hypergraph theory [11,16,22]. In this paper, we study spectral properties of circulant](https://static.documents.pub/doc/80x56/5f0fef1a7e708231d4469d18/circulant-tensors-with-applications-to-spectral-resonance-imaging-3102526-and.jpg)
