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A p p e n d i x BM A T R I X A L G E B R ABas ic Def in i t i onsA rectangular array of numbers or functions

a n &i2 ' ' a i nA = : : ... : t 5 " 1 )

I ^ m l ^ m 2 ' * ' Q"mn \is known as a matrix and is denoted in the text by capital boldface letters.The numbers or functions di j are the elements of the m atrix an d the subscripti denotes the row and j the column. A matr ix with m rows and n columnsis of order (m , n) or an m x n (m by n) matrix. If m = n, the matrix is asquare matrix.

If a matrix is an m x 1 m atrix , it is a column vector. If it is a 1 x nmatrix, it is a row vector. Generally, lower-case boldface letters are used todenote column or row vectors.A square matrix in which all elements are zero except those on the maindiagonal, an, a 2 2 r ianm is a diagonal matrix. If all elements of a diagonalmatrix are unity, then the matrix is the identity matrix and is denoted as I.When aij = aji, the matrix is called a symmetrical matrix. A null or zeromatrix is one in which all elements are zero.

A d d i t i o n a n d S u b t r a c t i o nTwo matrices can be added or subtracted only if they are of the same order.Thus , if A has elements a^ and B has elements 6y, then, if

481

Electromechanical Motion Devices, Second Editionby Paul Krause, Oleg Wasynczuk and Steven PekarekCopyright 2012 Institute of Electrical and Electronics Engineers, Inc.

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482 APPENDIX B

C has elements Q,-, where

or if

then

C = A + B (B-2)

Cij = aij + bi j ( B - 3 )

C = A - B (B-4)

C{j Q*ij vij \D-0)Also, addition is commutative,

A + B - B + A (B-6)and associative,

(A + B ) + C = A + (B + C) (B-7)Obviously,

A + O = A (B-8)where O is the zero matrix.M u l t i p l i c a t i o nIf the matrix A is multiplied by a scalar, every element of the matrix ismu ltiplied by th e scalar. For exam ple, kA means all elements of A aremultiplied by the constant k; tA means that all elements of A are multipliedby time t

To mu ltiply two m atrices, say A B , it is necessary tha t th e num ber ofcolum ns of A equ al th e num ber s of rows of B . If A is of order raxn and Bof order nxp, then the order of A B is mxp. The elements ofC = A B (B-9)

are obtained by multiplying the elements by the zth row of A by the corresponding elements of the jth column of B and adding these products. Inparticular,Cij = anbij + a i2b2j + . . . + a inbnj (B-10)

In (B-9), A is said to premultiply B, whereas B is said to postmultiplyA. Let

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MATRIX ALGEBRA 483

andB

Then

However,A B =

B A =

1 2 3 14 5 6 J

- 7 - 8 "9 100 - 1 1 _

I" 11 - 2 1[ 17 - 4 8

1-1

39 - 5 4 - 6 949 68 8744 - 5 5 --66

( B - l l )

(B-12)

(B-13)

(B-14)We see that, in general, matrix multiplication is not commutative; thus,

A B ^ B A (B-15)Multiplying a matrix of mxn by a column vector of n x l yields a columnvector of m x l . M ultiplication of a row vector of l x n and a column vectorof n x l yields a function (scalar) tha t is the sum of the p rod uct of specificelements of each vector.Multiplying the identity matrix by A yields A, that is,

A I - IA = A (B-16)We can show that

A{tHJ) = {Ati)*JA ( B + C ) = A B + A C( B + C ) A = B A + C A

inally, consider the simultaneous linear equationsbx + 3y-2z = 14

x + y Az = 7

[O-U)(B-18)(B-19)

(B-20)(B-21)

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484 APPENDIX B6x + 3z 1

Let us write the above equations in the formA x = b

HereA =

x

5 3 - 21 1 - 46 0 3xyz

b = 14- 71

(B-22)

(B-23)

(B-24)

(B-25)

(B-26)In this case, A is called the coefficient matrix. Care must be taken not toconfuse the column vector x and the variable x.T r a n s p o s eT he transpose of a matrix A is denoted as A T . Th e tran spo se of A isobtained by interchanging the rows and columns of A. Thus, if

A =

A T

1 2 3 45 6 7 8r i2

34

5 1678 J

(B-27)

(B-28)

The transpose possesses the following properties:( A T ) T = A

(A + B + C) T = A T + B r + C T( A B ) T = B T A T

(B-29)(B-30)(B-31)

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MATRIX ALGEBRA 485( A B C ) T = C T B r A a (B-32)

P a r t i t i o n i n gPartitioning of matrices is used throughout the text. It is helpful in matrixmultiplication. For example, let A and B be parti t ioned as

(B-33)A =B =

r c[ E[ G[ J

D 1F JH 1K

where C through K are subm atrices. T he produ ct of A B isC G + D J C H + D KE G + F J E H + F KB =

(B-34)

(B-35)D e t e r m i n a n t sEvery square matrix has a scalar associated with it called its determinantIn particular, if A is a square matrix, say,

A = on2131

122232

132333

(B-36)then the determinant of A is denoted det A or |A|

d e t A = 11 12 13021 22 2331 32 33

(B-37)It is important to note the difference between (B-36) and (B-37); (B-36) represents a matrix that is a square array of elements, whereas (B-37) representsa scalar associated with the matrix A.

The det A is determined by obtaining the minors and cofactors. Givena matrix A, a minor is the determinant of any square submatrix of A. Thecofactor of the element CLI j i s a scalar obtained by mu ltiplying (l)l+J t imesthe minor obtained from A by removing the zth row and j t h column. Tofind the determinant of the square matrix A:1. Pick any one row or any one column of the matrix.2. For each element in the row or column chosen, find its cofactor.

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486 APPENDIX B

3. M ultiply each elem ent in the row or colum n chosen by its cof actor andsum the results.

The sum is the determinant of the matrix. For example, find det A, whereA =

3 5 0-1 1 13 - 6 4 (B-38)Expanding in the second column,d e t A = ( 5 ) ( - l ) 1 + 2 - 1 13 4 + ( 1 ) ( - 1 ) 2 + 2

3 03 4 + ( - 6 ) ( - l ) 3 + 2

= (-5)(-7) + ( l )(12) + (6)(3) = 65

3 0- 1 1

(B-39)A d j o i n tT he adjoint matrix of a square matrix A, denoted adjoint A or A a , is formedby replacing each element a^ by th e cof acto r a^ and transposing. Thus, theadjoint of (B-36) is

adjoint A = an au #13a2l #22 #23#31 a 3 2 #33#11 #21 #31#12 #22 #32#13 #23 #33

(B-40)

InverseT he inverse of a square matrix A is written as A - 1 and is defined asA " X A = A A " 1 = I (B-41)

In the text, parentheses are used to avoid confusion with superscripts, that is,the inverse is denoted ( A ) - 1 . The inverse is defined only for square matrices.In particular,adjoint AA - * = det A (B-42)

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MATRIX ALGEBRA 487If det A is zero, A does not possess an inverse and is said to be singular.Consider a 2x2 matrix:

A =adjoint A =

a n &i2&21 a 2 2

&22 12- a 2 i a n _det A = ana 22 - ^12^21

Find the inverse of (B-38). The cofactor of an isa n = (3X-1)1*1 1 1-6 4 = (3) (10) = 30

(B-43)

(B-44)

(B-45)

(B-46)Finding the cofactor of each element and transposing yields the adjoint ofA. Thus ,

adjoint A = 30 20 1535 12 180 33 32 (B-47)

The det A is given by (B-39), hence,A - * = adjoint Adet A

165

" 30350

201233

15 "1832 (B-48)D e r i v a t i v e sThe derivative of the matrix A, denoted (d/dt) A or pA, is the derivativeof each element of the matrix. The derivative of A, given by (B-l), is

pA =pan pa12 pa2\ p a 2 2 pa nl pan2

pain pa2npann

(B-49)

where p is the operator d/dt.

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488 APPENDIX B

M a t r i x F o r m u l a t i o nIn the text, we deal with equations of the formA _ 1 y = A - V l x + p [ A _ 1 z] (B-50)

where A is a square, nonsingular matrix, the elements of which may befunctions of time, and p is the operator d/dt. Solving (B-50) for y is achievedby premultiplying by A. Thus,A A - 1 y = A A V l x + A p[A _ 1z ]

= r A A _ 1 x + A f p A - ^ z + A A _ 1 [pz] (B-51)Since A A _ 1 = I, (B-51) may be written as

y = r lx + A[pA _ 1 ]z + pz (B-52)