An Index to Matrices.pauli.pedersen

249

Pauli Pedersen: 12. An index to matrices

12. AN INDEX TO MATRICES

--- definitions, facts and rules ---

This index is based on the following goals and observations:

¯Togive the userquick reference to anactualmatrix definition or rule,

the index form is preferred. However, the index should to a large

extent be self-explaining.

¯The contents is selected in relation to the importance for matrix for-

mulations in solid mechanics.

¯The existence of good computer software for the numerical calcula-

tions, diminishes the need for details on specific procedures.

¯The existence of good computer software for the formulamanipula-

tions means that extended analytical work is possible.

¯The index is written by a non---mathematician (but hopefully with-

out errors), and is written for readers with a primary interest in

applying the matrix formulation without studying the matrix theory

itself.

¯Available chapters or appendices in books on solid mechanics are

not found extensive enough, and good classic books on linear alge-

bra are found too extensive. For further reference, see e.g.

250


Gantmacher, F.R. (1959) ‘The Theory of Matrices’,

Chelsea Publ. Co., Vol. I, 374 p., Vol. II, 276 p.

Gel’fand, I.M. (1961) ‘Lectures on Linear Algebra’,

Interscience Publ. Inc., 185 p.

Muir, T. (1928) ‘A Treatise on the Theory of Determinants’,

Dover Publ. Inc., 766 p.

Noble, B. and Daniel, I.W. (1988) ‘Applied Linear Algebra’,

Prentice---Hall, third ed., 521 p.

Strang, G. (1988) ‘Linear Algebra and its Applications’,

Harcourt Brace Jovanovich, 505 p.

Strang, G. (1986) ‘Introduction to Applied Mathematics’,

Wellesley---Cambridge Press, 758 p.

It will be noticed that the rather lengthy notation with [ ] for matrices

and { } for vectors (columnmatrices) is preferred for the more simple

boldface or underscore notations. The reason for this is that the reader

by the brackets is constantly remindedabout the fact thatwe aredealing

with a blockof quantities.Tomiss this point is catastrophic inmatrix cal-

culations. Furthermore, the lengthy notation adds to the possibilities

for direct graphical interpretation of the formulas.

Cross-reference in the index is symbolized by boldface writings. The

preliminary advices from colleagues and students are verymuch appre-

ciated, and I shall be grateful for further critics and comments that can

improve the index.

251


ADDITION Matrices are added by adding the corresponding elements

of matrices

[C]= [A]+ [B] with Cij= Aij+ Bij

The matrices must have the same order.

ANTI--METRIC or See skew---symmetric matrix.

ANTI--SYMMETRIC

matrix

BILINEAR FORM For a matrix [A] we define the bilinear form by

{X}T[A]{Y}

252


BILINEAR For a symmetric, positive definitematrix [A] we have by definition for

INEQUALITY the following two quadratic forms:

{Xa}T[A]{Xa

}= ua> 0 for {Xa}≠ {0}

{Xb}T[A]{X

b}= u

b> 0 for {X

b}≠ {0}

The bilinear form fulfills the inequality

{Xa}T[A]{X

b}≤ 1

2(ua+ u

b)

i.e. less than or equal to the mean value of the values of the quadratic

forms.

This follows directly from

�{Xa}T–{X

b}T�[A]�{Xa}–{Xb

}�≥ 0

and only equality for {Xa}= {Xb} . Expanding we get with the defini-

tions

ua+ ub–2{Xa}

T[A]{Xb}≥ 0

because [A]T= [A] .

253


BIORTHOGONALITY From the description of the generalized eigenvalue problem (see this)

conditions with right and left eigenvectors {Φ}iand {Ψ}

iwe have

{Ψ}Tj �[A] – λ

i[B]�{Φ}

i= 0

and

{Ψ}Tj�[A] – λ

j[B]�{Φ}i= 0

which by subtraction gives

(λi– λj)�{Ψ}T

j [B]{Φ}i�= 0

For different eigenvalues

λi≠ λj

this implies

{Ψ}Tj [B]{Φ}

i= 0

and thus also

{Ψ}Tj [A]{Φ}

i = 0

which is termed the biorthogonality conditions.

For a symmetric eigenvalue problem {Ψ}i=

{Φ}i (see orthogonality

conditions).

254


CHARACTERISTIC From the determinant condition

POLYNOMIUM

(generalized) |[A]λ2+ [B]λ+ [C]|= 0

with the square matrices [A] , [B] and [C] all of order n we obtain

a polynomium of order 2n in λ . This polynomium is termed the char-

acteristic polynomium of the triple ([A] , [B] , [C]).

Specific cases as

|[A]λ2+ [C]|= 0

|[I]λ+ [C]|= 0

are often encountered.

CHOLESKI See factorization of a matrix.

factorization /

triangularization

COEFFICIENTS See elements of a matrix.

of a matrix

COFACTOR The cofactor of a matrix element is the corresponding minor with an

of a matrix element appropriate sign. If the sum of row and column indices for the matrix

element is even, the cofactor is equal to theminor. If this sum is odd the

cofactor is the minor with reversed sign, i.e.

Cofactor (Aij)= (–1)i+j Minor (Aij)

255


COLUMN A column matrix is a matrix with only one column, i.e. order m× 1 .

matrix The notation { } is used for a columnmatrix. The name columnvector

or just vector is also used.

CONGRUENCE Acongruence transformation of a squarematrix [A] to a squarematrix

transformation [B] of the same order is by the regular transformation matrix [T] of

the same order

[B]= [T]T[A][T]

Matrices [A] and [B] are said to be congruent matrices, they have the

same rank and the same definiteness, but not necessarily same eigenva-

lues. A congruence transformation is also an equivalence transforma-

tion.

CONJUGATE The conjugate transpose is a transformation of matrices with complex

TRANSPOSE elements. Complex conjugate is denoted by a bar and transpose by a

superscript T . With a short notation (from the name Hermitian) we

denote the combined transformation as

[A]H= [A]

T

256


CONTRACTED For a symmetricmatrix, a simpler contracted notation in terms of a row

NOTATION or columnmatrix is possible.Of the notations which keep the orthogo---

for a symmetric matrix nal transformation, we choose the form with 2� ---factors multiplied to

the off diagonal elements in the matrix, i.e.

{B} from [A] with

Bi= A

iifor i= 1, 2, ..., n

Bn+...= 2� Aij for j> i

(The ordering within {B} symbolized by n+... is not specified).

257


CONVEX SPACE For a symmetric, positive definitematrix [A] we have by definition for

by positive the following two quadratic forms:

definite matrix

{Xa}T[A]{Xa

}= ua ; 0< ua

{Xb}T[A]{X

b}= u

b; 0< u

b≤ ua

The matrix [A] describes a convex space such that for

{Xα}= α{Xa}+ (1 – α){X

b} ; 0≤ α≤ 1

we have for all values of α

{Xα}T[A]{Xα

}= uα≤ ua

Inserting directly we have with [A]T

= [A]

�α{Xa}T+ (1 – α){X

b}T�[A]�α{Xa

}+ (1 – α){Xb}�

= α2{Xa

}T[A]{Xa}+ (1 – α)2{X

b}T[A]{X

b}+ 2α(1 – α){Xa

}T[A]{Xb}

= α2ua+ (1 – α)2u

b+ 2α(1 – α){Xa

}T[A]{Xb}

From the bilinear inequality we have

{Xa}T[A]{X

b}≤ 1

2(ua+ u

b)

and thus with ub≤ ua we can substitutive greater values and obtain

{Xα}T[A]{Xα

}≤ α2ua+ (1 – α)2ua+ 2α(1 – α)ua= ua

258


DEFINITENESS

For a symmetric matrix the notions of: are used if, for the matrix:

¯ positive definite ¯ all eigenvalues are positive

¯ positive semi---definite ¯ eigenvalues non---negative

¯ negative definite ¯ all eigenvalues are negative

¯ negative semi---definite ¯ eigenvalues non---positive

¯ indefinite ¯ both positive and negative eigenvalues

See specifically positive definite, negative definite and indefinite for

alternative statement of these conditions.

259


DETERMINANT The determinant of a squarematrix is a scalar, calculated as the sum of

of a matrix products of elements from the matrix. The symbol of two vertical lines

det ([A])= |[A]|

is used for this quantity.

For a square matrix of order two the determinant is

|[A]|=

A11

A21

A12

A22

=A

11A

22– A

12A

21

For a square matrix of order three the determinant is

|[A]|=

A11

A21

A31

A12

A22

A32

A13

A23

A33

=

A11A

22A

33+A

12A

23A

31+A

13A

21A

32– A

31A

22A

13– A

32A

23A

11– A

33A

21A

12

We note that for each product the number of elements is equal to the

order of the matrix, and that in each product a row or a column is only

represented by one element. Totally for a matrix of order n there are

n! terms to be summed.

For further calculation procedures see determinants by minors/cofac-

tors.

260


DETERMINANTS A determinant can be calculated in terms of cofactors (or minors), by

BY MINORS / expansion in terms of an arbitrary row or column.

COFACTORS

As an example, for a matrix of order three expansion of the third col-

umn yields:

A11

A21

A31

A12

A22

A32

A13

A23

A33

= A

13Minor(A

13) – A

23Minor(A

23)+A

33Minor(A

33)

See determinant of a matrix for direct comparison.

DETERMINANT The product of the determinants for a regular matrix [A] and its

OF AN INVERSE inverse [A]–1 is equal to 1

matrix

|[A]–1|= 1��|[A]|�

DETERMINANT The determinant of a product of squarematrices is equal to the product

OF A PRODUCT of the individual determinants, i.e.

of matrices

|[A][B]| = |[A]||[B]|

261


DETERMINANT The determinant of transposed square matrix is equal to the deter---

OF A TRANSPOSED minant of the matrix itself, i.e.

matrix

|[A]T| = |[A]|

DIAGONAL A diagonal matrix is a matrix where all off diagonal elements have the

matrix value zero

[A] a diagonal matrix when Aij = 0 for i ≠ j

and at least one diagonal element is non---zero. This definition also

holds for non---square matrices, as by singular value decomposition.

DIFFERENTIAL See functional matrix.

matrix

DIFFERENTIATION Differentiation of a matrix is carried out by differentiation of each

of a matrix element

[C]= d([A])�db with Cij= d(Aij)�db

DIMENSIONS See order of a matrix.

of a matrix

262


DOT PRODUCT See scalar product of two vectors.

of two vectors

DYADIC PRODUCT The dyadic product of two vectors {A} and {B} of the same order

of two vectors n results in a square matrix [C] of order n× n , but only with rank

1

[C]= {A}{B}T with Cij= A iBj

Dyadic products of vectors of different order can also be defined,

resulting in a matrix of order m× n .

EIGENPAIR The eigenpair λi , {Φ}i is a solution to an eigenvalue problem. The

eigenvector {Φ}icorresponds to the eigenvalue λ

i.

EIGENVALUES The eigenvalues λi

of a square matrix [A] are the solutions to the

of a matrix standard form for the eigenvalue problem, with

([A] – λi[I]){Φ}

i= {0}⇒ |[A] – λ

i[I]|= 0

which gives a characteristic polynomium.

263


EIGENVALUE With [A] and [B] being two squarematrices of order n , the general---

PROBLEM ized eigenvalue problem is defined by

�[A] – λi[B]�{Φ}

i= {0} for i= 1, 2, ..., n

or by

{Ψ}Ti�[A] – λ

i[B]�= {0}

Tfor i= 1, 2, ..., n

The pairs of eigenvalue, eigenvectors are λi, {Φ}

iand λ

i, {Ψ}T

i with

{Φ}ias right eigenvector and {Ψ}

ias lefteigenvector. Theeigenvalue

problem has n solutions with possibility for multiplicity.

With [B] being an identity matrix we have the standard form for an

eigenvalue problem, while for [B] not being an identity matrix the

name generalized eigenvalue problem is used.

EIGENVECTOR An eigenvector {Φ}iis the vector---part of a solution to an eigenvalue

problem. The word eigen reflects the fact that the vector is transformed

into itself except for a factor, the eigenvalue λi.

ELEMENTS The elements of amatrix [A] are the individual entries Aij . In amatrix

of a matrix of order m× n there are mn elements Aij , for i= 1, 2, ...,m ,

j= 1, 2, ..., n .Elements are also called themembersor the coefficients

of the matrix.

EQUALITY Twomatrices of the sameorder are equal if the corresponding elements

of matrices of each of the matrices are equal, i.e.

[A]= [B] if Aij= B ij for all ij

264


EQUIVALENCE An equivalence transformation of a matrix [A] to a matrix [B] (not

transformations necessarily square matrices) by the two square, regular transformation

matrices [T1] and [T

2] is

[B]= [T1][A][T2

]

Matrices [A] and [B] are said to be equivalent matrices and have the

same rank.

EXPONENTIAL The exponential of a square matrix [A] is defined by its power series

of a matrix expansion

e[A]t := [I]+ [A]t+ [A]2 t2

2!+ [A]

3 t3

3!+��

The series always converges, and the exponential properties are kept,

i.e.

e[A]te[A]s= e[A](t+s) , e[A]te[A](–t)

= [I] , d�e[A]t��dt= [A]e[A]t

265


FACTORIZATION A symmetric, regularmatrix [A] of order n can be factorized into the

of a matrix product of a lower triangular matrix [L] , a diagonalmatrix [B] and

the upper triangular matrix [L]T

all of the order n

[A]= [L][B][L]T

In a Gauss factorization the diagonal elements of [L] are all 1 .

A Choleski factorization is only possible for positive semi---definite

matrices, and then [B]= [I] and we get

[A]= [L][L]T

with Lii

not necessarily equal to 1 .

FROBENIUS The Frobenius norm of a matrix [A] is defined as the square root of

norm of a matrix the sum of the squares of all the elements of [A] .

For a square matrix of order 2 we get

Frobenius= A2

11+ A2

22+A2

12+ A2

21�

and thus for a symmetricmatrix equal to the squareroot of the invariant

I3.

For a square matrix of order 3 we get

Frobenius= �(A2

11+A2

21+A2

31)+ (A2

22+A2

12+A2

32)+ (A2

33+A2

13+ A2

23)�½

and thus for a symmetricmatrix equal to the squareroot of the invariant

I4.

266


FULL RANK See rank of a matrix.

FUNCTIONAL The functional matrix [G] consists of partial derivatives --- the partial

MATRIX derivatives of the elements of a vector {A} of order m with respect

to the elements of a vector {B} of order n . Thus the functionalmatrix

is of the order m× n

[G]=∂{A}

∂{B}with G

ij=∂Ai

∂Bj

The name gradient matrix is also used. A square functional matrix is

named a Jacobimatrix, and the determinant of this matrix as the Jaco-

bian.

GAUSS See factorization of a matrix.

factorization /

triangularization

GENERALIZED See eigenvalue problem.

EIGENVALUE

PROBLEM

GEOMETRIC A vector of order two or three in an Euclidian plane or space. See vec---

vector tors. By a geometric vector we mean a oriented piece of a line (an

“arrow”).

267


GRADIENT See functional matrix.

matrix

HERMITIAN A square matrix [A] is termed Hermitian if it is not changed by the

matrix conjugate transpose transformation, i.e.

[A]H= [A]

Every eigenvalue of aHermitianmatrix is real, and the eigenvectors are

mutually orthogonal, as for symmetric real matrices.

HESSIAN AHessian matrix [H] is a square, symmetricmatrix containing second

matrix order derivatives of a scalar F with respect to the vector {A}

[H]= ∂2F∂{A}∂{A}

with Hij =∂2F

∂A i∂A j

268


HURWITZ The Hurwitz determinants up to order eight are defined by

determinants

Hi:=

a1

a0

a3

a2

a1

a0

a5

a4

a3

a2

a1

a0

a7

a6

a5

a4

a3

a2

a1

a0

a8

a7

a6

a5

a4

a3

a2

a8

a7

a6

a5

a4

a8

a7

a6

a8

to be read in the sense that Hiis the determinant of order i defined

in the upper left corner (principal submatrix). More specifically,

H1= a

1

H2= a

1a2– a

0a3

H3= H

2a3– (a

1a4– a

0a5)a

1

··

If the highest order is n , then am = 0 for m> n , and therefore the

highest Hurwitz determinant is given by

Hn= Hn–1

an

269


IDENTITY An identity matrix [I] is a square matrix where all diagonal elements

matrix have the value one and all off diagonal elements have the value zero

[I] := [A] with Aii= 1, A

ij= 0 for i≠ j

The name unit matrix is also used for the identity matrix.

INDEFINITE Asquare, realmatrix [A] is called indefinite if positive aswell asnega---

matrix tive values of {X}T[A]{X} exist, i.e.

{X}T[A]{X} ><

0

depending on the actual vector (column matrix) {X} .

INTEGRATION The integral of a matrix is the integral of each element

of a matrix

[C]= �[A]dx with Cij=�Aijdx

INVARIANTS For matrices which transforms by similarity transformations we can

of similar matrices determine a number of invariants, i.e. scalars which do not change by

the transformation. The number of independent invariants are equal to

the order of the matrix, and as any combination is also an invariant

many different forms are possible. Tomention some important invaria-

nts we have eigenvalues, trace, determinant, and Frobenius norm. The

principal invariants are the coefficients of the characteristic polyno-

mium.

270


INVARIANTS For the square, symmetric matrix [A] of order 2 we have

of symmetric, similar

matrices of order 2

[A]= �A11

A12

A12

A22

�with invariants being the trace I

1by

I1= A

11+A

22

and the determinant I2by

I2= A

11A22

– A2

12

Taking as an alternative invariant I3by

I3= (I

1)2 – 2I

2= A2

11+A2

22+ 2A2

12

we get the squared length of the vector {A} contracted from [A] by

{A}T= {A11, A

22, 2� A

12}

Setting up the polynomium to find the eigenvalues of [A] we find

λ2 – I

1λ+ I

2= 0

and again see the importance of the invariants I1and I

2, termed the

principal invariants.

271


INVARIANTS For the square, symmetric matrix [A] of order 3 we have

of symmetric, similar

matrices of order 3

[A]=

A11

A12

A13

A12

A22

A23

A13

A23

A33

with invariants being the trace I1by

I1= A

11+A

22+A

33

the norm I2by

I2= �A

11A

22– A2

12�+ �A

22A

33– A2

23�+ �A

11A

33– A2

13�

and the determinant I3by

I3= |[A]|

These three invariants are the principal invariants and they give the

characteristic polynomium by

λ3– I

1λ2+ I

2λ – I

3= 0

The squared length of the vector {A} contracted from [A] by

{A}T= �A11, A

22, A

33, 2� A

12, 2� A

13, 2� A

23�

isI4= A2

11+A2

22+A2

33+ 2A2

12+ 2A2

13+ 2A2

23

related to the principal invariants by

I4= (I

1)2 – 2I

2

and therefore another invariant, equal to the squared Frobenius norm.

272


INVERSE The inverse of a square, regularmatrix is the square matrix, where the

of a matrix product of the two matrices is the identity matrix. The notation [ ]–1

is used for the inverse

[A]–1[A]= [A][A]

–1= [I]

INVERSE OF A From the matrix product in partitioned form

PARTITIONED

matrix

[A]

[C]

[B]

[D]

[E]

[G]

[F]

[H]=

[I]

[0]

[0]

[I]

follows the four matrix equations

[A][E]+ [B][G]= [I] ; [A][F]+ [B][H]= [0]

[C][E]+ [D][G]= [0] ; [C][F]+ [D][H]= [I]

Solving these we obtain (in two alternative forms)

[H]= [D]–1

– [D]–1

[C][F]

[G]= – [D]–1

[C][E]

[E]= �[A] – [B][D]–1[C]�–1

[F]= – [E][B][D]–1

[E]= [A]–1

– [A]–1

[B][G]

[F]= – [A]–1

[B][H]

[H]= �[D] – [C][A]–1[B]�–1

[G]= – [H][C][A]–1

The special case of an upper triangular matrix, i.e. [C]= [0] gives

[H]= [D]–1

[G]= [0]

[E]= [A]–1

[F]= – [A]–1

[B][D]–1

[E]= [A]–1

[F]= – [A]–1

[B][D]–1

[H]= [D]–1

[G]= [0]

273


The special case of a symmetric matrix, i.e. [C]= [B]T

gives

[H]= [D]–1

– [D]–1

[B]T

[F]

[G]= – [D]–1

[B]T

[E]

[E]= �[A] – [B][D]–1[B]T�–1

[F]= – [E][B][D]–1

= [G]T

[E]= [A]–1

– [A]–1

[B][G]

[F]= – [A]–1

[B][H]

[H]= �[D] – [B]T[A]–1[B]�–1

[G]= – [H][B]T

[A]–1

= [F]T

The matrices to be inverted, are assumed to be regular.

INVERSE OF The inverse of a product of square, regular matrices is the product of

A PRODUCT the inverse of the individual multipliers, but in reverse sequence

([A][B])–1= [B]

–1[A]

–1

It follows directly from

([B]–1[A]

–1)([A][B])= [I]

INVERSE OF The inverse of a matrix of order two is given by

ORDER TWO

�A11A

12

A21

A22

�–1

= � A22

–A12

–A21

A11

� 1|[A]|

with the determinant given by

|[A]|= A11A

22– A

21A

12

274


INVERSE OF The inverse of a matrix of order three is given by

ORDER THREE

A11

A12

A13

A21

A22

A23

A31

A32

A33

–1

=

(A22A

33– A

32A

23) , (A

32A

13– A

12A

33) , (A

12A

23– A

22A

13)

(A31A

23– A

21A

33) , (A

11A

33– A

31A

13) , (A

21A

13– A

11A

23)

(A21A

32– A

31A

22) , (A

31A

12– A

11A

32) , (A

11A

22– A

21A

12)

1|[A]|

With the determinant given by

|[A]|=

A11A

22A

33+A

12A

23A

31+A

13A

21A

32– A

31A

22A

13– A

32A

23A

11– A

33A

21A

12

INVERSE OF The inverse and the transpose transformations can be interchanged

TRANSPOSED

matrix

([A]T)–1= ([A]–1)

T= [A]

–T

from which follows the definition of the symbol [ ]–T

.

JACOBI The Jacobi matrix [J] is a square functional matrix. We define it here

matrix as thematrix containing the derivatives of the elements of a vector {A}

with respect to the elements of a vector {B} , both of order n

[J]=∂{A}

∂{B}with J

ij =∂A i

∂Bj

275


JACOBIAN The Jacobian J is the determinant of the Jacobi matrix, i.e.

determinant

J= |[J]|

and thus a scalar.

JORDAN BLOCKS A Jordan block is a square upper---triangular matrix of order equal to

themultiplicity of an eigenvalue with a single corresponding eigenvec-

tor. All diagonal elements are theeigenvalue andall theelements of the

first upper codiagonal are 1 . Remaining elements are zero. Thus the

Jordan block [Jλ] of order 3 corresponding to the eigenvalue λ is

[Jλ]=

λ

00

1λ

0

01λ

Multiple eigenvalues with linear independent eigenvectors belongs to

different Jordan blocks.

Jordan blocks or order 1 aremost common, as this results for eigenva-

lue problems described by symmetric matrices.

JORDAN FORM The Jordan form of a square matrix [A] is the similar matrix [J] con-

sisting of Jordan blocks along the diagonal (block diagonal), and with

remaining elements equal to zero.

Only when we havemultiple eigenvalues with a single eigenvector will

the Jordan form be different from pure diagonal form. Jordan forms

represent the closest---to---diagonal outcome of a similarity transfor-

mation.

276


LAPLACIAN See determinants by minors/cofactors.

EXPANSION

of determinants

LEFT The left eigenvector {Ψ}T (row matrix) corresponding to eigenvalue

eigenvector λiis defined by

{Ψ}T

i ([A] – λi[B])= {0}

T

see eigenvalue problem.

LENGTH The length |{A}| of a vector is the square---root of the scalar product

of a vector of the vector with itself

|{A}|= {A}T{A}�

A geometric vector has an invariant length, but this do not hold for all

algebraic vector definitions.

LINEAR Consider a matrix [A] of order m× n , constituting the n vectors

DEPENDENCE / {A}ifor i= 1, 2, ..., n . Then if there exist a non---zero vector {B} of

LINEAR order n such that

INDEPENDENCE

[A]{B}= [{A}1{A}

2��{A}

n]{B}= {0}

then the vectors {A}iare said to be linear dependent. The vector {B}

contains a set of linear combination factors.

If on the other hand

[A]{B}= {0} only for {B}= {0}

277


then the vectors {A}iare said to be linear independent.

278


MEMBERS See elements of a matrix.

of a matrix

MINOR The minor of a matrix element is a determinant, i.e. a scalar.

of a matrix element

The actual squarematrix corresponding to this determinant is obtained

by omitting the row and column corresponding to the actual element.

Thus, for a matrix of order 3, the minor corresponding to element A12

become

Minor(A12)=

�A21

A23

A31

A33

�= A21A

33– A

31A

23

MODAL The modal matrix corresponding to an eigenvalue problem is a square

matrix matrix constituting all the linear independent eigenvectors

[Φ]= [{Φ}1{Φ}

2��{Φ}

n]

and the generalized eigenvalue problem can then be stated as

[A][Φ] – [B][Φ][Γ] = [0]

Note that the diagonalmatrix [Γ] of eigenvaluesmust be post---multi-

plied.

279


MULTIPLICATION The product of two matrices is a matrix, where the resulting element

of two matrices �ij� is the scalar productof the i---th rowof the first matrixwith the j---th

column of the second matrix

[C] = [A][B] with Cij=�K

k=1

AikBkj

The number of columns in the first matrix must be equal to the number

of rows in the second matrix (here K) .

MULTIPLICATION A matrix is multiplied by a scalar by multiplying each element by the

BY SCALAR scalar

[C]= b[A] with Cij= bAij

MULTIPLICITY In eigenvalue problems the same eigenvalue may be a multiple solu---

OF EIGENVALUES tion, mostly (but not always) corresponding to linear independent

eigenvectors.As an example a bimodal solution is a solution, where two

eigenvectors correspond to the same eigenvalue. Multiplicity of eigen-

values is also named algebraic multiplicity.

For non---symmetric eigenvalue problems multiple eigenvalues may

correspond to the same eigenvector. We then talk about, e.g., a double

eigenvalue/eigenvector solution (by contrast to a bimodal solution,

where only the eigenvalue is the same). Thismultiplicity is described by

the geometric multiplicity of the eigenvalue. For a specific eigenvalue

we have

1≤ geometric multiplicity≤ algebraic multiplicity

Note that the geometric multiplicity of an eigenvalue counts the number of linear independent

eigenvectors for this eigenvalue, and not the number of times that the eigenvector is a solution.

280


NEGATIVE DEFINITE A square, real matrix [A] is called negative or negative definite if for

matrix any non---zero vector (column matrix) {X} we have

{X}T[A]{X}< 0

The matrix is called negative semi---definite if

{X}T[A]{X}≤ 0

NORMALIZATION Eigenvectors can be multiplied with an arbitrary constant (even a

of a vector complex constant). Thus we have the possibility for a convenient scal-

ing, and often we choose the weighted norm. Here we scale the vector

{A}ito the normalized vector {Φ}

i

{Φ}i= {A}

i� {A}T

i [B]{A}i

�

by which we obtain

{Φ}Ti [B]{Φ}

i= 1

Alternative normalizations are by other norms, such as the 2---norm

{Φ}i = {A}

i� {A}Ti{A}�

i

or by the ∞ ---norm

{Φ}i = {A}

i�(Max|A j|)

281


NULL A null matrix (symbolized [0]) is a matrix where all elements have the

matrix value zero

[0] := [A] with Aij= 0 for all ij

A null matrix is also called a zero matrix. The null vector is a special

case.

ONE A one matrix (symbolized [1]) is a matrix where all elements have the

matrix value one

[1] := [A] with Aij= 1 for all ij

The one vector is a special case. Note the contrast to the identity (unit)

matrix [I] , which is a diagonal matrix.

ORDER The order of a matrix is the (number of rows)×(number of columns) .

of a matrix Usually the letters m× n are used, and a rowmatrix thenhas theorder

1× n while a column matrix has the order m× 1 . For square

matrices a single number gives the order. The order of a matrix is also

called the dimensions or the size of the matrix.

282


ORTHOGONALITY For an eigenvalue problem ([A] – λi[B]){Φ}

i= {0} with symmetric

conditions matrices [A] and [B] the biorthogonality conditions simplifies to

{Φ}Tj [B]{Φ}i= 0 , {Φ}

Tj [A]{Φ}i= 0

for non---equal eigenvalues, i.e. λi ≠ λj .

For standard form eigenvalue problems with [A] symmetric this fur-

ther simplifies to

{Φ}Tj {Φ}

i= 0 , {Φ}Tj [A]{Φ}

i= 0 for λi≠ λj

Using normalization of the eigenvectors we can obtain

{Φ}Ti [B]{Φ}

i= 1 or {Φ}Ti {Φ}

i= 1

and thus

{Φ}Ti [A]{Φ}

i= λi

Orthogonal, normalized eigenvectors are termed orthonormal.

283


ORTHOGONAL An orthogonal transformation of a square matrix [A] to a square

transformations matrix [B] of the same order is by the orthogonal transformation

matrix

[T]–1= [T]

T

and thus the transformation is both a congruence transformation and

a similarity transformation

[B]= [T]T[A][T]= [T]–1[A][T]

Matrices [A] and [B] are said to be orthogonal similar, and have same

rank, same eigenvalues, same trace and same determinant (same

invariants).

If matrix [A] is symmetric, matrix [B] is also symmetric, which do not

hold generally for similar matrices.

ORTHONORMAL A orthonormal set of vectors {X}i fulfill the conditions

{X}Ti [A]{X}

j= �01 for

for

i≠ j

i= j

284


PARTITIONING Partitioning ofmatrices is a very important tool to get closer insight and

of matrices overview. By the example

[A]=[A]

11[A]

12

[A]21

[A]22

we see that the submatrices are given indices exactly like thematrix ele-

ments themselves.

Multiplication on submatrix level is identical to multiplication on ele-

ment level. For example see inverse of a partitioned matrix.

POSITIVE DEFINITE Asquare, realmatrix [A] is called positive or positive definite if for any

matrix non---zero vector (column matrix) {X} we have

{X}T[A]{X}> 0

The matrix is called positive semi---definite if

{X}T[A]{X}≥ 0

285


POSITIVE DEFINITE The conditions for a square matrix [A] to be positive definite can be

matrix conditions stated in many alternative forms. From the Routh---Hurwitz---Lie-

nard---Chipart teoremwe can directly in termsofHurwitzdeterminants

obtain the necessary and sufficient conditions for eigenvalues with pos-

itive real part.

For a matrix of order 2 we get that

[A]= �A11

A21

A12

A22

� has positive real part of all eigenvalues if and only if

(A11+ A

22)> 0 and A

11A

22– A

12A

21> 0

and the conditions for a symmetricmatrix (A21= A

12) to be positive

definite is then

A11> 0 , A

22> 0 and A

11A

22– A2

12> 0

For a matrix of order 3 we get that

[A]=

A11

A21

A31

A12

A22

A32

A13

A23

A33

has positive real part of all eigenvalues if and only if

I1= (A

11+A

22+A

33)> 0

I2= �(A

11A

22– A

21A

12)+ (A

22A

33– A

32A

23)+ (A

11A

33– A

31A

13)�> 0

I3= |[A]|> 0 and I

1I2– I

3> 0

and the conditions for a symmetric matrix to be positive definite will

then be

A11> 0 , A

22> 0 , A

33> 0

286


A11A

22– A2

12> 0 , A

22A33

– A2

23> 0 , A

11A

33– A2

13> 0 , |[A]|> 0

287


POSITIVE DEFINITE Assume that the two square, real matrices [A] and [B] of the same

SUM order are positive definite, then their sum is also positive definite.

of matrices Using the symbol � for positive definite, we have

[A]� 0 , [B]� 0 ⇒ ([A]+ [B])� 0

It follows directly from the definition

{X}T([A]+ [B]){X}= {X}T[A]{X}+ {X}T[B]{X}> 0

because both terms are positive for {X}≠ {0} .

From this also follows directly that

�α[A]+ (1 – α)[B]�� 0 for 0≤ α≤ 1

which implies that [A]� 0 is a convex condition.

Identical relations hold for negative definite matrices.

POWER The power of a square matrix [A] is symbolized by

of a matrix

[A]p= [A][A]�� [A] (p times)

[A]–p = [A]–1[A]–1�� [A]

–1(p times)

[A]0= [I] ; [A]

p[A]

r= [A]

(p+r); �[A]

p�r = [A]pr

288


PRINCIPAL Theprincipal invariants are the coefficients of the characteristic poly---

INVARIANTS nomium for similar matrices.

PRINCIPAL The principal submatrices of the squarematrix [A] of order n , are the

SUBMATRIX n squared matrices of order k (1≤ k≤ n) found in the upper left

corner of [A] .

PRODUCT See multiplication of two matrices.

of two matrices

PRODUCTS Three different products of vectors are defined. The scalar product or

of two vectors dot product resulting in a scalar. The vector product or cross product

resulting in a vector, and especially used for vectors of order three.

Finally, the dyadic product resulting in a matrix.

PROJECTION A projection matrix different from the identity matrix [I] is a square

matrix singular matrix that is unchanged when multiplied by itself

[P][P]= [P] , [P]–1

non–existent

289


PSEUDOINVERSE The pseudoinverse [A+] of a rectangular matrix [A] of order m× n

of a matrix always exists. When [A] is a regular matrix the pseudoinverse is the

same as the inverse. Given the singular value decomposition of [A] by

[A]= [T1][B][T2

]T

then with the diagonal matrix [C] of order n×m defined from the

diagonal matrix [B] of order m× n by

[C] from C ii= 1�Bii for B ii≠ 0 (other C ij= 0)

the pseudoinverse [A+] is given by the product

[A+]= [T2][C][T1

]T

Case 1: [A] is a n×m matrix where n> m . The solution to

[A]{X}= {B} with the objective of minimizing the error

�{e}T{e} , {e}= [A]�X�− {B}� , is given by

�X�= �[A]T[A]�

−1

[A]T{B}

Case 2: [A] is a n×m matrix where n< m . The solution to

[A]{X}= {B} with the objective of minimizing the length of the solu-

tion ��X�T�X�� , is given by

�X�= [A]

T�[A][A]T�−1

{B}

290


QUADRATIC By a symmetricmatrix [A] of order n we define the associated qua---

FORM dratic form

{X}T[A]{X}

that gives a homogeneous, second order polynomial in the n parame-

ters constituting the vector {X} . The quadratic form is used in many

applications, and thus knowledge about its transformations, definite-

ness etc. is of vital importance.

RANK The rank of a matrix is equal to the number of linearly independent

of a matrix rows (or columns) of the matrix. The rank is not changed by the trans-

pose transformation.

From a matrix [A] of order (m× n) we can, by omitting a number

of rows and/or a number of columns, get square matrices of any order

from 1 to theminimum of m,n . Normally there will be several differ-

ent matrices of each order.

The rank r is defined by the largest order of these square matrices, for

which the determinant is non---zero, i.e. the order of the “largest” regu-

lar matrix we can extract from [A] .

Only a zero matrix has the rank 0 .

The rank of any other matrix will be

1≤ r≤ min (m, n)

If r = min(m,n) we say that the matrix has full rank.

291


REAL With [A] and [B] being two real and symmetricmatrices, then for the

EIGENVALUES eigenvalue problem

([A] – λi[B]){Φ}

i= {0}

¯ if λiis complex, then {Φ}

iis also complex ([A] and [B] regular)

¯ if λi,{Φ}

iis a complex pair of solution, then the complex conju-

gated pair λi,{Φ}

iis also a solution.

The condition derived under biorthogonality conditions for these two

pairs is

(λi– λ

i)({Φ}T

i[B]{Φ}

i)= 0

which expressed in real and imaginary parts are

2 Im(λi)�Re({Φ}Ti )[B] Re({Φ}i)+ Im({Φ}

T

i )[B] Im({Φ}i)�= 0

It now follows that if [B] is a positive definitematrix, then Im(λi)= 0

and we have real eigenvalues.

REGULAR A non---singular matrix, see singular matrix.

matrix

RIGHT The right eigenvector {Φ}i(column matrix) corresponding to eigen---

eigenvector values λi

is defined by

([A] – λi[B]){Φ}

i= {0}

292



293


ROTATIONAL For two dimensional problems we shall list some important orthogonal

transformation transformation matrices. The elements of these matrices involves

matrices trigonometric functions of the angle θ defined in the figure. For short

notation we also define

θ

c1= cosθ s

1= sin θ

c2= cos 2θ s

2= sin 2θ

c4= cos 4θ s

4= sin 4θ

The two Cartesian coordinate systems with the definition of the angle θ .

We then have for rotation of a geometric vector {V} of order 2

{V}y =

[Γ]{V}x

with [Γ]= � c1–s1

,,s1c1� ; [Γ]

–1= [Γ]

T

For a symmetric matrix [A] of order 2× 2 , contracted with the

2� ---factor to the vector {A}T= {A11, A22, 2� A12} we have

{A}y= [T]{A}x

294


with [T]= 12

1+ c2

1 – c2

– 2� s2

,

,

,

1 – c2

1+ c2

2� s2

,

,

,

2� s2

– 2� s2

2c2

; [T]

–1= [T]

T

For a symmetricmatrix [B] of order 3× 3 , contractedwith the 2� ---

factor to the vector {B}T = {B11, B22, B33, 2� B12, 2� B13, 2� B23} we

have

{B}y = [R]{B}x

with [R]–1= [R]

Tand [R]= 1

8·

3+ 4c2+ c

4

3 – 4c2+ c

4

2 – 2c4

2� – 2� c4

– 4s2– 2s

4

– 4s2+ 2s

4

,

,

,

,

,

,

3 – 4c2+ c

4

3+ 4c2+ c

4

2 – 2c4

2� – 2� c4

4s2– 2s

4

4s2+ 2s

4

,

,

,

,

,

,

2 – 2c4

2 – 2c4

4+ 4c4

– 2 2� + 2 2� c4

4s4

– 4s4

,

,

,

,

,

,

2� – 2� c4

2� – 2� c4

– 2 2� + 2 2� c4

6+ 2c4

2 2� s4

– 2 2� s4

,

,

,

,

,

,

4s2+ 2s

4

– 4s2+ 2s

4

– 4s4

– 2 2� s4

4c2+ 4c

4

4c2– 4c

4

,

,

,

,

,

,

4s2– 2s

4

– 4s2– 2s

4

4s4

2 2� s4

4c2– 4c

4

4c2+ 4c

4

Note that the listed orthogonal transformation matrices [Γ] , [T] and

[R] only refer to two dimensional problems, where the rotation is spe-

cified by a single parameter (the angle θ) .

ROW Arowmatrix is amatrixwith only one row, i.e.order 1× n . Thenota---

matrix tion { }T is used for a row matrix ( { } for column matrix and T for

transposed). The name row---vector or just vector is also used.

295


SCALAR PRODUCT The scalar product of two vectors {A} and {B} of the same order n

of two vectors results in a scalar C

(standard

Euclidean norm) C= {A}T{B}=�n

i=1

AiB

i

The scalar product is also called the dot product.

SCALAR PRODUCT The scalar product of two complex vectors {A} and {B} of the same

of two complex vectors order n involves the conjugate transpose transformation

(standard norm)

C= {A}H{B}=�n

i=1

�Re(Ai) – i Im(A

i)��Re(B

i)+ i Im(B

i)�

With this definition the length of a complex vector {A} is obtained by

|{A}|2= {A}H{A}=�

n

i=1

��Re(Ai)�

2

+ �Im(Ai)�

2�

SIMILARITY A similarity transformation of a square matrix [A] to a square matrix

transformations [B] of the same order is by the regular transformation matrix [T] of

the same order

[B]= [T]–1[A][T]

Matrices [A] and [B] are said to be similar matrices, they have the

same rank and the same eigenvalues, i.e. the same invariants, but dif-

ferent eigenvectors, related by [T] . A similarity transformation is also

an equivalence transformation.

296


SINGULAR A singular matrix is a square matrix for which the corresponding

matrix determinant has the value zero, i.e.

[A] is singular if |[A]|= 0 , i.e. [A]–1 does not exist

If not singular, the matrix is called regular or non---singular.

SINGULAR VALUE Any matrix [A] of order m× n can be factorized into the product of

DECOMPOSITION an orthogonal matrix [T1] of order m , a rectangular, diagonalmatrix

[B] of order m× n and an orthogonal matrix [T2]T

of order n

[A]= [T1][B][T

2]T

The r singular values (positive values) on the diagonal of [B] are the

square roots of the non---zero eigenvalues of both [A][A]T

and

[A]T[A] ,and the columns of [T

1] are the eigenvectors of [A][A]

Tand

the columns of [T2] are the eigenvectors of [A]

T[A] .

SIZE See order of a matrix.

of a matrix

297


SKEW A skew matrix is a specific skew symmetric matrix of order 3, defined

matrix to have a more workable notation for the vector product of two vectors

of order 3 . From the vector {A} the corresponding skew matrix is

defined by

[A~

]=

0

A3

–A2

–A3

0

A1

A2

–A1

0

by which {A}× {B}= [A~

]{B} .

The tilde superscript is normally used to indicate this specific matrix.

From {B}× {A}= – {A}× {B} follows

[B~

]{A}= – [A~

]{B}

SKEW SYMMETRIC A square matrix is termed skew---symmetric if the transposed trans---

matrix formation only changes the sign of the matrix

[A]T= – [A] , i.e. Aji= – Aij for all ij (Aii= 0)

The skew symmetric part of a square matrix [B] is obtained by the dif-

ference 12([B]–[B]

T) .

298


SPECTRAL For a symmetric matrix a spectral decomposition is possible. The

DECOMPOSITION eigenvalues λiof the matrix [A] are factors in this decomposition

of a symmetric matrix

[A]=�n

i=1

λi[B]

i=�

n

i=1

λi{Φ}

i{Φ}T

i

where {Φ}i

is the eigenvector corresponding to λi

(orthonormal

eigenvectors).

SQUARE A square matrix is a matrix where the number of rows equals to the

matrix number of columns, thus the order of the matrix is n× n or simply

n .

STANDARD FORM The standard form for an eigenvalue problem is

for eigenvalue problem

[A]{Φ}i= λ

i{Φ}

i

or

{Ψ}Ti[A]= λ

i{Ψ}T

i


SUBTRACTION Matrices are subtracted by subtracting the corresponding elements

of matrices

[C]= [A] – [B] with Cij= A ij – Bij

The matrices must have the same order.

299


SYMMETRIC With [A] and [B] being two symmetric matrices of order n , the left

EIGENVALUE eigenvectors will be equal to the right eigenvectors. From the descrip---

PROBLEM tion of eigenvalue problem this means

{Ψ}i= {Φ}

i

and thus the biorthogonality conditions simplifies to the orthogonality

conditions. The symmetric eigenvalue problem have only real eigenva-

lues and real eigenvectors.

SYMMETRIC A square matrix is termed symmetric if the transposed transformation

matrix does not change the matrix

[A]T= [A] , i.e. A ji= Aij for all ij

The symmetric part of a square matrix [B] is obtained by the sum12([B]+ [B]

T) .

TRACE The trace of a square matrix [A] of order n is the sum of the diagonal

of a square matrix elements

trace([A])=�n

i=1

Aii

300


TRANSFORMATION The different transformations like equivalence, congruence, similarity

matrices and orthogonal are characterized by the involved square, regular trans-

formation matrices. The equivalence transformation of

[B]= [T1][A][T2

]

is a congruence transformation if [T1]= [T2

]T

and it is a similarity

transformation if [T1]= [T2

]–1

. The orthogonal transformation,

which at the same time is a congruence and a similarity transformation,

thus assumes [T1]= [T2

]T= [T2

]–1

.

TRANSPOSE The transposed of a matrix is the matrix with interchanged rows/

of a matrix columns. The superscript T is used as notation for this transformation

[B]= [A]T

with Bij= Aji for all ij

The transposed of a row matrix is a column matrix, and vise versa.

The transposed matrix of a transposed matrix is the matrix itself

([AT])T= [A]

TRANSPOSE The transposed of a product of matrices is the product of the trans---

OF A PRODUCT posed of the individual multipliers, but in reverse sequence

([A] [B])T = [B]T[A]

T

It follows directly from

301


Cij=�K

k=1

AikBkj and Cji=�K

k=1

AjkBki =�K

k=1

BkiAjk

TRIANGULAR A triangularmatrix is a squarematrix with only zeros above the diago---

matrix nal (lower triangular matrix)

[L] with Lij= 0 for j> i

or below the diagonal (upper triangular matrix)

[U] with Uij= 0 for j< i

TRIANGULARIZA-- See factorization of a matrix.

TION

of a matrix

UNIT See identity matrix.

matrix

VECTORS As a common name for row matrices and column matrices, the name

vector is used.

Some authors distinguish between geometric vectors (oriented piece of

a line) of order twoor three andalgebraic vectors.Algebraic vectors are

column matrices and row matrices of any order.

302


VECTOR PRODUCT The vector product of two vectors {A} and {B} , both of the order 3

of two vectors is a vector {C} defined by

{C}= {A}× {B} with

C1

C2

C3

=

A2B3

A3B1

A1B2

–

–

–

A3B2

A1B3

A2B1

The vector product is also called the cross product. See skewmatrix for

an easier notation.

ZERO See null matrix.

matrix

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