dk3162app8
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rotordynamicsTRANSCRIPT
APPENDIX 8
Matrix Calculation Review
A matrix A (n�m) is rectangular array containing n rows and m columns:
Aðn�mÞ ¼
a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .. . . . . . . . . . . .am1 am2 . . . amn
266664
377775 ¼ ½aij�, i ¼ 1, . . . ,m, n ¼ 1, . . . ,n
A transposed matrix ATðn�mÞ is the matrix, in which rows are interchanged with the
columns:
ATðn�mÞ ¼ Aðm� nÞ
Two matrices of the same order n�m can be added and/or subtracted, by adding and/orsubtracting individual elements:
Aðn�mÞ þ Bðn�mÞ ¼ ½aij þ bij�
Two matrices can be multiplied only if the number of columns, ‘‘m’’ of the first matrix A(n�m) equals to the number of rows of the second matrix B (m� p). The product is thematrix of the order (n� p):
Aðn�mÞ � Bðm� pÞ ¼ Cðn� pÞ
The elements of the matrix C are corresponding sums of products:
½ckl� ¼Xmj¼1
akjbjp
" #, k ¼ 1, . . . ,n, l ¼ 1, . . . p
EXAMPLE 1
a11 a12 a13
a21 a22 a23
" #�
b11 b12
b21 b22
b31 b32
2664
3775 ¼
a11b11 þ a12b21 þ a13b31 a11b12 þ a12b22 þ a13b32
a21b11 þ a22b21 þ a23b31 a21b12 þ a22b22 þ a23b32
" #
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© 2005 by Taylor & Francis Group, LLC
EXAMPLE 2
1 2
3 4
" #x
y
" #¼
xþ 2y
3xþ 4y
" #
Generally, the matrices in the product cannot be interchanged, because AB 6¼ BA.A product of a number and a matrix is a matrix, in which each element is multiplied by
this number:
�A ¼ �½aij� ¼ ½�aij�
EXAMPLE 3
51 2
3 4
� �¼
5 10
15 20
� �
Two matrices are equal, when all their corresponding elements are equal:
Aðn�mÞ ¼ Bðn�mÞ, aij ¼ bij, i ¼ 1, . . . ,m, j ¼ 1, . . . ,n
Cofactor of the matrix ½aij� is:
Cof aij ¼ ðminorÞð�1Þiþj
EXAMPLE 4
Cof
a b c
d f g
h i j
2664
3775 ¼
fj� gi gh� dj di� hf
ic� bj aj� ch bh� ai
bg� cf ag� cd af� bd
2664
3775
EXAMPLE 5
Cof ¼1 2
3 4
� �¼
4 �3
2 1
� �
The adjoint matrix is the transpose of the cofactor matrix:
adj A ¼ ðcof AÞT
EXAMPLE 6
adj ¼1 �23 4
� �¼
4 2�3 1
� �
Determinant, det A can be calculated for any square matrix A, that is, when n¼m.
1020 ROTORDYNAMICS
© 2005 by Taylor & Francis Group, LLC
EXAMPLE 7
det
a b c
d e f
h i j
2664
3775 ¼
a b c
d e f
h i j
��������
�������� ¼ aðfj� igÞ þ bðgh� djÞ þ cðdi� fhÞ
Diagonal matrix is the square matrix, which has nonzero elements on the main diagonaland the off-diagonal elements equal zero:
a 0 0 . . . 0
0 b 0 . . . 0
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
0 0 0 . . . z
2666666664
3777777775
The unit diagonal matrix, I, has all diagonal elements equal one:
I ¼
1 0 0 . . . 0
0 1 0 . . . 0
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
0 0 0 . . . 1
2666666664
3777777775
The inverse matrix, A�1 of the matrix A is the matrix, which multiplied by A (from theleft or from the right) gives in product the unit matrix:
AA�1¼ I, A�1A ¼ I
The inverse matrix is as follows:
A�1¼
adjA
detA
EXAMPLE 8
A ¼a b
c d
" #, det A ¼ ad� bc, adj A ¼
d �b
�c a
" #, A�1
¼1
ad� bc
d �b
�c a
" #
A�1A ¼1
ad� bc
d �b
�c a
" #a b
c d
" #¼
1
ad� bc
da� bc db� bd
�acþ ac �bcþ ad
" #¼
1 0
0 1
" #
The matrix equation
AX ¼ B
MATRIX CALCULATION REVIEW 1021
© 2005 by Taylor & Francis Group, LLC
can be solved for X if the matrix A is square. Multiplying both sides of this equation fromthe left by the inverse matrix A�1 this equation results as follows:
A�1AX ¼ A�1B
IX ¼ A�1B
X ¼ A�1B
EXAMPLE 9
Ax1x2
� �¼
72
� �where A ¼
2 13 �2
� �
Calculate inverse matrix A:
A�1¼
1
� 4� 3
�2 �1�3 2
� �¼
1
7
2 13 �2
� �
x1x2
� �¼ A�1 7
2
� �¼
1
7
2 13 �2
� �72
� �¼
1
7
14þ 221� 4
� �¼
16=717=7
� �
Thus x1 ¼ 16=7, x2 ¼ 17=7:
1022 ROTORDYNAMICS
© 2005 by Taylor & Francis Group, LLC