Research Article A Note on the Square Roots of a Class of...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 601243, 6 pageshttp://dx.doi.org/10.1155/2013/601243

Research ArticleA Note on the Square Roots of a Class of Circulant Matrices

Ying Zhang, Huisheng Zhang, and Guoyan Chen

Department of Mathematics, Dalian Maritime University, Dalian 116026, China

Correspondence should be addressed to Ying Zhang; [email protected]

Received 28 July 2013; Accepted 3 November 2013

Academic Editor: Frank Werner

Copyright © 2013 Ying Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove that any 𝑘-circulantmatrix and any even order skew 𝑘-circulantmatrix are diagonalizable for any 𝑘 ∈ C.Then, we proposetwo algorithms for computing the square roots of the 𝑘-circulant matrix and the skew 𝑘-circulant matrix, respectively. In particular,we show that the square roots of the 𝑘-circulant matrix are still 𝑘-circulant matrices. Both the theoretical analysis and the numericalexperiments show that our algorithms are faster than the standard Schur method.

1. Introduction

Given a matrix 𝐴, a matrix 𝑋 is called a square root of 𝐴 if𝑋2= 𝐴. Matrix square roots appear in a variety of branches

of mathematics, such as Markov models of finance, the solu-tion of differential equations, the computation of the polardecomposition, and the matrix sign function [1]. A numberof methods have been proposed for computing the squareroots of a matrix [2–9]. Among them, the Schur method [7]is the most popular and becomes the standard method forcomputing the matrix square roots. However, Schur methodis not so efficient when the matrix order is relatively high.Thus, it is very desirable to design fast computing methodswhich can make full use of the particular properties of thematrices when the matrices possess special structures.

Circulant matrices and their generalizations have a widerange of applications in signal processing, coding theory,digital image disposal, self-regress design, Toeplitz systems,and so on [10–14]. A relatively comprehensive survey aboutcirculant matrices can be found in [15]. Recently, Lu andGu [16] presented two efficient algorithms to compute thesquare roots of circulant matrices and quasi-skew circulantmatrices, respectively. As they are based on LL iteration [17]and themodified Schulz iterativemethod, the two algorithmsare faster than the standard Schur method. Subsequently,Mei generalized those methods and presented algorithms tocompute the square roots of 𝑘-circulant matrices and skew𝑘-circulant matrices [18]. These algorithms are also fasterthan the standard Schur algorithm, but the work is restricted

to the case in which the matrix is of even order and 𝑘 ∈ R

and can not be directly extended to compute the 𝑝th root.In this paper, we first show that 𝑘-circulant matrices of

any order for any complex number 𝑘 are diagonalizable anddevelop an algorithm to compute their principal square roots.Then, we show that skew 𝑘-circulant matrices of even orderfor any complex number 𝑘 are diagonalizable, and developan algorithm to compute their principle square roots. Both ofour algorithms are theoretically and experimentally provedto be faster than the standard Schur method. Compared withthe work in [18], our methods are more general in that theyare valid for any 𝑘-circulant matrix and any even order skew𝑘-circulant matrix, where 𝑘 can be any complex number.Theremainder of this paper is organized as follows. In Section2, we compute the square roots of 𝑘-circulant matrices. InSection 3, we compute the square roots of skew 𝑘-circulantmatrices. In Section 4, we present two numerical experimentsto exhibit the efficiency of the proposed algorithms in termsof the CPU time.

2. Square Roots of 𝑘-Circulant Matrices

An 𝑛 × 𝑛 complex matrix

Circ𝑘(𝑎) :=

[[[[[[

[

𝑎0

𝑎1

𝑎2

⋅ ⋅ ⋅ 𝑎𝑛−1

𝑘𝑎𝑛−1

𝑎0

𝑎1

. . . 𝑎𝑛−2

𝑘𝑎𝑛−2

𝑘𝑎𝑛−1

𝑎0

⋅ ⋅ ⋅ 𝑎𝑛−3

......

... d...

𝑘𝑎1

𝑘𝑎2

𝑘𝑎3⋅ ⋅ ⋅ 𝑎

0

]]]]]]

]

(1)

2 Journal of Applied Mathematics

is called a 𝑘-circulant matrix, where 𝑎 = (𝑎0, 𝑎1, . . . , 𝑎

𝑛−1) ∈

C𝑛 and 𝑘 ∈ C. In particular, 1-circulant matrices are circulantmatrices, and −1-circulant matrices are skew circulant matri-ces [19].

Another equivalent definition of a 𝑘-circulantmatrix is asfollows [18]: let C𝑛×𝑛 be the set of all 𝑛 × 𝑛 complex matrices,and then, 𝐴 ∈ C𝑛×𝑛 is a 𝑘-circulant matrix if and only if 𝐴 =

𝐺−1𝐴𝐺, where 𝐺 = Circ

𝑘([0, 1, 0, . . . , 0]). In this section, we

show that 𝑘-circulant matrices are diagonalizable.

Lemma 1 (see [20]). If 𝐴 and 𝐵 are two 𝑘-circulant matricesof the same order, then 𝐴 + 𝐵 is also a 𝑘-circulant matrix.

Lemma 2 (see [20]). If 𝐴 is a 𝑘-circulant matrix, then for any𝑙 ∈ C, 𝑙𝐴 is also a 𝑘-circulant matrix.

Lemma 3 (see [20]). Let 𝐺 = Circ𝑘([0, 1, 0, . . . , 0]); then,

𝐺𝑖=

[[[[[[[[[[[

[

0 ⋅ ⋅ ⋅ 0 1 0 ⋅ ⋅ ⋅ 0

0 ⋅ ⋅ ⋅ 0 0 1 ⋅ ⋅ ⋅ 0

... d...

...... d

...0 ⋅ ⋅ ⋅ 0 0 0 ⋅ ⋅ ⋅ 1

𝑘 ⋅ ⋅ ⋅ 0 0 0 ⋅ ⋅ ⋅ 0

... d...

...... d

...0 ⋅ ⋅ ⋅ 𝑘 0 0 ⋅ ⋅ ⋅ 0

]]]]]]]]]]]

]

(2)

is also a 𝑘-circulant matrix. In particular, 𝐺𝑛 = 𝑘𝐸, where 𝐸 isthe identity matrix.

Lemma 4 (see [20]). Let 𝐺 = Circ𝑘([0, 1, 0, . . . , 0]) and let

𝑘 = 𝑟𝑒𝑖𝜃 (𝑟 ≥ 0, 0 ≤ 𝜃 ≤ 2𝜋); then, the eigenvalues of 𝐺 are

𝜃𝑗= 𝑟1/𝑛𝑒𝑖(𝜃+2𝜋(𝑗−1)/𝑛)

, (𝑗 = 1, . . . , 𝑛) , (3)

where 𝑖 is the imaginary unit.

Theorem 5 (see [20]). The matrix 𝐶 = Circ𝑘(𝑎) is a 𝑘-

circulant matrix of the form (1) if and only if 𝐶 can berepresented by

𝐶 =

𝑛−1

∑

𝑗=0

𝑎𝑗𝐺𝑗, (4)

where 𝑎 = (𝑎0, . . . , 𝑎

𝑛−1), 𝐺 = Circ

𝑘([0, 1, 0, . . . , 0]), and 𝐺0 =

𝐸.

Theorem 6 (see [20]). Let 𝐶 = Circ𝑘(𝑎) be a 𝑘-circulant

matrix; then, the eigenvalues of𝐶 are 𝜆𝑗= 𝜙(𝜃

𝑗) (𝑗 = 1, . . . , 𝑛),

where 𝑎 = (𝑎0, . . . , 𝑎

𝑛−1), 𝜙(𝑥) = 𝑎

0+𝑎1𝑥+𝑎2𝑥2+⋅ ⋅ ⋅+𝑎

𝑛−1𝑥𝑛−1,

and 𝜃𝑗is defined by (3).

Theorem 7 (see [20]). Any 𝑘-circulant matrix Circ𝑘(𝑎) can be

diagonalized as follows:

𝑉−1⋅ Circ𝑘(𝑎) ⋅ 𝑉 = diag [𝜙 (𝜃

1) , 𝜙 (𝜃

2) , . . . , 𝜙 (𝜃

𝑛)] (5)

with the matrix

𝑉 =

[[[[[[

[

1 1 1 ⋅ ⋅ ⋅ 1

𝜃1

𝜃2

𝜃3

⋅ ⋅ ⋅ 𝜃𝑛

𝜃2

1𝜃2

2𝜃2

3⋅ ⋅ ⋅ 𝜃

2

𝑛

......

... d...

𝜃𝑛−1

1𝜃𝑛−1

2𝜃𝑛−1

3⋅ ⋅ ⋅ 𝜃𝑛−1

𝑛

]]]]]]

]

, (6)

where 𝜃𝑗(𝑗 = 1, . . . , 𝑛) is defined as (3).

By (5), we can easily obtain the following result.

Corollary 8. The square roots of 𝑘-circulant matrix Circ𝑘(𝑎)

are as follows:

𝑉 diag [𝜙(𝜃1)1/2

, 𝜙(𝜃2)1/2

, . . . , 𝜙(𝜃𝑛)1/2

]𝑉−1. (7)

Remark 9. We mention that the diagonalization methods[19–22] were used to compute the square roots of 𝑘-circulantmatrices, where 𝑘 is restricted to 1 or −1. However, ourdiagonalization method is valid for any 𝑘 ∈ C.

Next, we show that the square roots of a 𝑘-circulantmatrix are still 𝑘-circulant matrices.

Theorem 10. Let 𝑘 = 𝑟𝑒𝑖𝜃 (𝑟 ≥ 0, 0 ≤ 𝜃 ≤ 2𝜋), let 𝜃

𝑗=

𝑟1/𝑛𝑒𝑖(𝜃+2𝜋(𝑗−1)/𝑛) (𝑗 = 1, . . . , 𝑛), and let𝑉 be defined as (6). For

any diagonal matrixΛ = diag(𝜆1, . . . , 𝜆

𝑛) ∈ C𝑛×𝑛,𝑉Λ𝑉−1 is a

𝑘-circulant matrix Circ𝑘([𝑏0, . . . , 𝑏

𝑛−1]), where (𝑏

0, . . . , 𝑏

𝑛−1) =

(𝜆1, . . . , 𝜆

𝑛)𝑉−1.

Proof. ByTheorem 7, whether𝑉Λ𝑉−1 is a 𝑘-circulant matrixdepends on whether the following system of linear equationswith unknown vector (𝑏

0, . . . , 𝑏

𝑛−1)

𝑏0+ 𝑏1𝜃1+ 𝑏2𝜃2

1+ ⋅ ⋅ ⋅ + 𝑏

𝑛−1𝜃𝑛−1

1= 𝜆1,

𝑏0+ 𝑏1𝜃2+ 𝑏2𝜃2

2+ ⋅ ⋅ ⋅ + 𝑏


2= 𝜆2

...

𝑏0+ 𝑏1𝜃𝑛+ 𝑏2𝜃2

𝑛+ ⋅ ⋅ ⋅ + 𝑏


𝑛= 𝜆𝑛

(8)

is consistent. Obviously the coefficient matrix of (8) isinvertible. Thus, we have

(𝑏0, . . . , 𝑏

𝑛−1) = (𝜆

1, . . . , 𝜆

𝑛) 𝑉−1. (9)

That is to say, 𝑉Λ𝑉−1 is a 𝑘-circulant matrix with the form ofCirc𝑘([𝑏0, . . . , 𝑏

𝑛−1]).

Remark 11. Theorem 10 provides a method to construct𝑘-circulant matrix with given eigenvalues. Obviously, thesquare roots with the form of (7) are 𝑘-circulant matricesCirc𝑘(𝑏), where 𝑏 = (𝜙(𝜃

1)1/2, . . . , 𝜙(𝜃

𝑛)1/2)𝑉−1.

Based onTheorem 7 and Corollary 8, we give the follow-ing algorithm for computing the principal square root of a𝑘-circulant matrix.

Journal of Applied Mathematics 3

Algorithm 12. Compute a principal square root of a 𝑘-circulant matrix 𝐴 ∈ C𝑛×𝑛.

Step 1. Compute the eigenvalues 𝜙(𝜃𝑗) = ∑

𝑛−1

𝑠=0𝑎𝑠(𝜃𝑗)𝑠 (𝑗 =

1, . . . , 𝑛) of 𝐴.

Step 2. Compute 𝜙(𝜃𝑗)1/2 (𝑗 = 1, . . . , 𝑛) such that

arg(𝜙(𝜃𝑗)1/2) (𝑗 = 1, . . . , 𝑛) ∈ (−𝜋/2, 𝜋/2).

Step 3. Compute the inverse of 𝑉.

Step 4. Compute diag[𝜙(𝜃1)1/2, 𝜙(𝜃2)1/2, . . . , 𝜙(𝜃

𝑛)1/2]𝑉−1.

Step 5. Compute the square root 𝑉 diag[𝜙(𝜃1)1/2, 𝜙(𝜃2)1/2,

. . . , 𝜙(𝜃𝑛)1/2]𝑉−1 of 𝐴.

Then, we obtain 𝐵 = √𝐴 = Circ𝑘([𝑏0, . . . , 𝑏

𝑛−1]).

The cost of Step 1 is about 𝑂(𝑛 log 𝑛) flops by discreteFourier transform [18]. The cost of Step 2 is 𝑂(𝑛) flops.The cost of Step 3 is about 𝑂(𝑛 log2𝑛) flops [23]. The costof Step 4 is about 𝑂(𝑛2) flops. The cost of Step 5 is about𝑂(𝑛2log2𝑛) flops [24]. So, it needs about 𝑂(𝑛2log2𝑛) flops

in total. The algorithm has the same complexity as thediagonalization methods in [19–21]. But the methods thereinare only concerned with the case that 𝑘 = 1 and 𝑘 = −1.If we use the Schur method, it needs about 𝑂(𝑛3) flops intotal [7]. We also mention a related work in [18], which onlyneeds about 𝑂(𝑛 log 𝑛) flops to compute the primary squareroot of a 𝑘-circulant matrix. However, that work restricts thematrix to be of even order and 𝑘 to be of real number. Thoserestrictions are not needed in our algorithm.

3. Square Roots of Skew 𝑘-Circulant Matrices

Let 𝑛 be an even number; then,𝐴 ∈ C𝑛×𝑛 is a skew 𝑘-circulantmatrix if 𝐴 = −𝐺

−1𝐴𝐺, where 𝐺 = Circ

𝑘([0, 1, 0, . . . , 0]) (see

[18]). Let 𝑘 = 𝑟𝑒𝑖𝜃 (𝑟 ≥ 0, 0 ≤ 𝜃 ≤ 2𝜋) and let

𝐽 =

[[[[[[[[[[

[

1 0 0 0 ⋅ ⋅ ⋅ 0 0

0 −1 0 0 ⋅ ⋅ ⋅ 0 0

0 0 1 0 ⋅ ⋅ ⋅ 0 0

0 0 0 −1 ⋅ ⋅ ⋅ 0 0

......

...... d

......

0 0 0 0 ⋅ ⋅ ⋅ 1 0

0 0 0 0 ⋅ ⋅ ⋅ 0 −1

]]]]]]]]]]

]

. (10)

Lemma 13. Let 𝐺 = Circ𝑘([0, 1, 0, . . . , 0]). Then,

(i) 𝐽𝐺𝐽 = −𝐺;(ii) 𝐽𝐺𝑗 = −𝐺

𝑗𝐽 for odd number 𝑗; 𝐽𝐺𝑗 = 𝐺

𝑗𝐽 for even

number 𝑗.

Lemma 14. Let

𝑉 = [V1, V2, . . . , V

2𝑚] =

[[[[[[

[

1 1 ⋅ ⋅ ⋅ 1

𝜃1

𝜃2

⋅ ⋅ ⋅ 𝜃2𝑚

𝜃2

1𝜃2

2⋅ ⋅ ⋅ 𝜃

2

2𝑚

...... d

...𝜃2𝑚−1

1𝜃2𝑚−1

2⋅ ⋅ ⋅ 𝜃2𝑚−1

2𝑚

]]]]]]

]

, (11)

where 𝜃𝑗= 𝑟1/2𝑚

𝑒𝑖(𝜃+2𝜋(𝑗−1)/2𝑚) (𝑗 = 1, . . . , 2𝑚); then, 𝐽V

𝑡=

V𝑚+𝑡

(𝑡 = 1, . . . , 𝑚).

Proof. Theproof is obvious by paying attention to the fact that𝜃𝑚+𝑡

= 𝜃𝑡𝑒𝑖𝜋.

Lemma 15. A skew 𝑘-circulant matrix 𝐴 of order 2𝑚 can bewritten in the form of

𝐽Circ𝑘([𝑎0, 𝑎1, . . . , 𝑎

2𝑚−1]) . (12)

Proof. Write 𝑛 = 2𝑚 and𝐴 = (𝑎𝑖𝑗)𝑛×𝑛

. Since𝐴 = −𝐺−1𝐴𝐺 we

have 𝐺𝐴 = −𝐴𝐺; that is,

[[[[[[

[

𝑎21

𝑎22

𝑎23

⋅ ⋅ ⋅ 𝑎2𝑛

𝑎31

𝑎32

𝑎33

⋅ ⋅ ⋅ 𝑎3𝑛

......

... d...

𝑎𝑛1

𝑎𝑛2

𝑎𝑛3

⋅ ⋅ ⋅ 𝑎𝑛𝑛

𝑘𝑎11

𝑘𝑎12

𝑘𝑎13

⋅ ⋅ ⋅ 𝑘𝑎1𝑛

]]]]]]

]

=

[[[[[[

[

−𝑘𝑎1𝑛

−𝑎11

−𝑎12

⋅ ⋅ ⋅ −𝑎1(𝑛−1)

−𝑘𝑎2𝑛

−𝑎21

−𝑎22

⋅ ⋅ ⋅ −𝑎2(𝑛−1)

......

... d...

−𝑘𝑎(𝑛−1)𝑛

−𝑎(𝑛−1)1

−𝑎(𝑛−1)2

⋅ ⋅ ⋅ −𝑎(𝑛−1)(𝑛−1)

−𝑘𝑎𝑛𝑛

−𝑎𝑛1

−𝑎𝑛2

⋅ ⋅ ⋅ −𝑎𝑛(𝑛−1)

]]]]]]

]

.

(13)

Then,

𝐴 =

[[[[[[

[

𝑎11

𝑎12

𝑎13

⋅ ⋅ ⋅ 𝑎1𝑛

−𝑘𝑎1𝑛

−𝑎11

−𝑎12

⋅ ⋅ ⋅ −𝑎1(𝑛−1)

−𝑘𝑎2𝑛

−𝑎21

−𝑎22

⋅ ⋅ ⋅ −𝑎2(𝑛−1)

......

... d...

−𝑘𝑎(𝑛−1)𝑛

−𝑎(𝑛−1)1

−𝑎(𝑛−1)2

⋅ ⋅ ⋅ −𝑎(𝑛−1)(𝑛−1)

]]]]]]

]

(𝑘𝑎11, 𝑘𝑎12, 𝑘𝑎13, . . . , 𝑘𝑎

1𝑛)

= (−𝑘𝑎𝑛𝑛, −𝑎𝑛1, −𝑎𝑛2, . . . , −𝑎

𝑛(𝑛−1)) .

(14)

Thus, we have that

𝐴 =

[[[[[[

[

𝑎11

𝑎12

𝑎13

⋅ ⋅ ⋅ 𝑎1𝑛

−𝑘𝑎1𝑛

−𝑎11

−𝑎12

⋅ ⋅ ⋅ −𝑎1(𝑛−1)

𝑘𝑎1(𝑛−1)

𝑘𝑎1𝑛

𝑎11

⋅ ⋅ ⋅ 𝑎1(𝑛−2)

......

... d...

−𝑘𝑎12

−𝑘𝑎13

−𝑘𝑎14

⋅ ⋅ ⋅ −𝑎11

]]]]]]

]

= 𝐽Circ𝑘([𝑎11, 𝑎12, . . . , 𝑎

1𝑛]) .

(15)

Theorem 16. A skew 𝑘-circulant matrix 𝐽Circ𝑘([𝑎0, . . . ,

𝑎2𝑚−1

]) of order 𝑛 = 2𝑚 is diagonalizable.

Proof. Write 𝐶 = Circ𝑘([𝑎0, . . . , 𝑎

2𝑚−1]); then 𝐶 = 𝑎

0𝐺0+

𝑎1𝐺 + 𝑎2𝐺2+ ⋅ ⋅ ⋅ + 𝑎

2𝑚−1𝐺2𝑚−1. By Lemma 13,

𝐽𝐶 = (𝑎0𝐺0− 𝑎1𝐺 + 𝑎2𝐺2− ⋅ ⋅ ⋅ − 𝑎

2𝑚−1𝐺2𝑚−1

) 𝐽. (16)

Thus, we have

𝐶 = 𝐽 (𝑎0𝐺0− 𝑎1𝐺 + 𝑎2𝐺2− ⋅ ⋅ ⋅ − 𝑎

2𝑚−1𝐺2𝑚−1

) 𝐽. (17)


100 200 300 400 500 600 700 800 900 1000 1100

Order of the matrix

CPU

tim

e (s)

102

101

100

10−1

10−2

Standard Schur methodAlgorithm 12

Figure 1: Comparison of the CPU time for Algorithm 12 and thestandard Schur method in logarithmic scale.

Standard Schur method

102

101

100

10−1

100 200 300 400 500 600 700 800 900 1000 1100

Order of the matrix

CPU

tim

e (s)

Algorithm 18

Figure 2: Comparison of the CPU time for Algorithm 18 and thestandard Schur method in logarithmic scale.

By Theorem 7, there exists an invertible matrix 𝑉 defined inLemma 14 such that

𝑉−1𝐽 (𝑎0𝐺0− 𝑎1𝐺 + 𝑎2𝐺2− ⋅ ⋅ ⋅ − 𝑎

2𝑚−1𝐺2𝑚−1

) 𝐽𝑉

= diag [𝜙 (𝜃1) , 𝜙 (𝜃

2) , . . . , 𝜙 (𝜃

2𝑚)] ,

(18)

where 𝜙(𝑥) = 𝑎0+ 𝑎1𝑥 + 𝑎2𝑥2+ ⋅ ⋅ ⋅ + 𝑎

2𝑚−1𝑥2𝑚−1. Then,

(𝑎0𝐺0− 𝑎1𝐺 + 𝑎2𝐺2− ⋅ ⋅ ⋅ − 𝑎

2𝑚−1𝐺2𝑚−1

) 𝐽𝑉

= 𝐽𝑉 diag [𝜙 (𝜃1) , 𝜙 (𝜃

2) , . . . , 𝜙 (𝜃

2𝑚)] .

(19)

By (16), we have

𝐽𝐶 [V1, V2, . . . , V

2𝑚]

= [𝐽V1, 𝐽V2, . . . , 𝐽V

2𝑚] diag [𝜙 (𝜃

1) , 𝜙 (𝜃

2) , . . . , 𝜙 (𝜃

2𝑚)] .

(20)

In order to obtain the eigenvalues and eigenvectors of 𝐽𝐶, for𝑡 = 1, . . . , 𝑚, we assume that there exist 𝑘

𝑡, 𝑘𝑚+𝑡

, and 𝑙 ∈ C,such that

𝐽𝐶 (𝑘𝑡V𝑡+ 𝑘𝑚+𝑡

V𝑚+𝑡

) = 𝑙 (𝑘𝑡V𝑡+ 𝑘𝑚+𝑡

V𝑚+𝑡

) . (21)

Combining this with (20) gives

𝑘𝑡𝜙 (𝜃𝑡) 𝐽V𝑡+ 𝑘𝑚+𝑡

𝜙 (𝜃𝑚+𝑡

) 𝐽V𝑚+𝑡

= 𝑙𝑘𝑡V𝑡+ 𝑙𝑘𝑚+𝑡

V𝑚+𝑡

. (22)

Now, we use Lemma 14 to get

𝑘𝑡𝜙 (𝜃𝑡) V𝑚+𝑡

+ 𝑘𝑚+𝑡

𝜙 (𝜃𝑚+𝑡

) V𝑡= 𝑙𝑘𝑚+𝑡

V𝑚+𝑡

+ 𝑙𝑘𝑡V𝑡. (23)

Since V𝑡and V𝑚+𝑡

are linearly independent, we have

𝑘𝑡𝜙 (𝜃𝑡) = 𝑙𝑘

𝑚+𝑡,

𝑘𝑚+𝑡

𝜙 (𝜃𝑚+𝑡

) = 𝑙𝑘𝑡.

(24)

So, 𝑙 = ±√𝜙(𝜃𝑡)𝜙(𝜃𝑚+𝑡

), 𝑘𝑚+𝑡

/𝑘𝑡= ±√𝜙(𝜃

𝑡)/𝜙(𝜃𝑚+𝑡

). Namely,

𝐽𝐶(√𝜙 (𝜃𝑚+𝑡

)V𝑡+ √𝜙 (𝜃

𝑡)V𝑚+𝑡

)

= √𝜙 (𝜃𝑡) 𝜙 (𝜃𝑚+𝑡

) (√𝜙 (𝜃𝑚+𝑡

)V𝑡+ √𝜙 (𝜃

𝑡)V𝑚+𝑡

) ,

𝐽𝐶 (√𝜙 (𝜃𝑚+𝑡

)V𝑡− √𝜙 (𝜃

𝑡)V𝑚+𝑡

)

= −√𝜙 (𝜃𝑡) 𝜙 (𝜃𝑚+𝑡

) (√𝜙 (𝜃𝑚+𝑡

)V𝑡− √𝜙 (𝜃

𝑡)V𝑚+𝑡

) .

(25)

Let

𝑊 = [√𝜙 (𝜃𝑚+1

)V1+ √𝜙 (𝜃

1)V𝑚+1

,

√𝜙 (𝜃𝑚+1

)V1− √𝜙 (𝜃

1)V𝑚+1

, . . . ,

√𝜙 (𝜃2𝑚)V𝑚+ √𝜙 (𝜃

𝑚)V2𝑚,

√𝜙 (𝜃2𝑚)V𝑚− √𝜙 (𝜃

𝑚)V2𝑚] .

(26)

Journal of Applied Mathematics 5

Then

𝐽𝐶𝑊 = 𝑊

[[[[[[[[[

[

√𝜙 (𝜃1) 𝜙 (𝜃𝑚+1

)

−√𝜙 (𝜃1) 𝜙 (𝜃𝑚+1

)

d

√𝜙 (𝜃𝑚) 𝜙 (𝜃2𝑚)

−√𝜙 (𝜃𝑚) 𝜙 (𝜃2𝑚)

]]]]]]]]]

]

. (27)

So,

𝑊−1𝐽𝐶𝑊 =

[[[[[[[[[

[

√𝜙 (𝜃1) 𝜙 (𝜃𝑚+1

)

−√𝜙 (𝜃1) 𝜙 (𝜃𝑚+1

)

d

√𝜙 (𝜃𝑚) 𝜙 (𝜃2𝑚)

−√𝜙 (𝜃𝑚) 𝜙 (𝜃2𝑚)

]]]]]]]]]

]

. (28)

Corollary 17. The square roots of an even order skew 𝑘-circulant matrix 𝐽Circ

𝑘(𝑎) are as follows:

𝑊

[[[[[

[

√𝜙(𝜃1)𝜙(𝜃𝑚+1

)

−√𝜙(𝜃1)𝜙(𝜃𝑚+1

)

d√𝜙(𝜃𝑚)𝜙(𝜃2𝑚)

−√𝜙(𝜃𝑚)𝜙(𝜃2𝑚)

]]]]]

]

1/2

𝑊−1. (29)

Proof. This is a direct result from (28).

Based on Theorem 16 and Corollary 17, we give thefollowing algorithm for computing the principal square rootof a skew 𝑘-circulant matrix.

Algorithm 18. Compute a principal square root of a skew 𝑘-circulant matrix 𝐴 ∈ C2𝑚×2𝑚.

Step 1. Compute 𝜙(𝜃𝑗) = ∑

2𝑚−1

𝑠=0𝑎𝑠(𝜃𝑗)𝑠 (𝑗 = 1, . . . , 2𝑚).

Step 2. Compute the eigenvalues of 𝐴: √𝜙(𝜃1)𝜙(𝜃𝑚+1

),

−√𝜙(𝜃1)𝜙(𝜃𝑚+1

), . . . , √𝜙(𝜃𝑚)𝜙(𝜃2𝑚), −√𝜙(𝜃

𝑚)𝜙(𝜃2𝑚).

Step 3. Compute the square roots of √𝜙(𝜃1)𝜙(𝜃𝑚+1

),

−√𝜙(𝜃1)𝜙(𝜃𝑚+1

), . . . , √𝜙(𝜃𝑚)𝜙(𝜃2𝑚), −√𝜙(𝜃

𝑚)𝜙(𝜃2𝑚) whose

arguments should be in (−𝜋/2, 𝜋/2).

Step 4. Compute (29).

Let 𝑛 = 2𝑚. The cost of Step 1 is about 𝑂(𝑛 log 𝑛) flopsby discrete Fourier transform [18]. The cost of Step 2 is 𝑂(𝑛).The cost of Step 3 is𝑂(𝑛). Since thematrix multiplication andinversion are equivalent in computational complexity [25],the cost of Step 4 is about 𝑂(𝑛2.4) [26]. So, it needs about𝑂(𝑛2.4) flops in total. If we use the Schur method [7] or the

method in [18], it needs about𝑂(𝑛3) flops in total. Moreover,our method allows 𝑘 to be a complex number, whereas themethod in [18] only permits 𝑘 to be a real number.

4. Numerical Experiments

We present numerical experiments to compare the algo-rithms presented in this paper and the standard Schurmethod with respect to the execution time. The code wasimplemented in MATLAB 7.0 and run on a Windows-basedmachine with 2GB of RAM and Intel Pentium Core DuoCPU running at 2.8GHz.

We gradually increase the order of thematrix by 100 from100 to 1100 and record the corresponding CPU time. Figure 1shows the execution time for Algorithm 12 and the standard


Schur method. Figure 2 compares the execution time forAlgorithm 18 and the standard Schur method. From thoseresults, we can confirm that our algorithms are clearly fasterthan the standard Schur method for computing the squareroots of 𝑘-circulant matrices and skew 𝑘-circulant mat-rices.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Thiswork is partly supported by theNational Natural ScienceFoundation of China (61101228) and by the FundamentalResearch Funds for the Central Universities (3132013337).

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