2005/8 Matrices-1
Matrices
2005/8 Matrices-2
A Matrix over a Field F (R or C)
nm
mnmmm
n
n
n
ij M
aaaa
aaaa
aaaa
aaaa
aA
][
nm321
3333231
2232221
1131211
m rows
n columns
size: m×n
ij-entry: aij F (ij-component)
2005/8 Matrices-3
The i-row (vector)
The j-column (vector)
),,,( 21 iniii aaaA
mj
j
j
j
a
a
a
A2
1
(row matrix)
(column matrix)
Square matrix: m = n
2005/8 Matrices-4
Diagonal matrix
),,,( 21 nddddiagA nn
n
M
d
d
d
00
00
00
2
1
Tracennija A ][
nnaaaATr 2211)(
unit nn matrix = diag(1, 1, , 1) zero matrix = diag(0, 0, , 0)
2005/8 Matrices-5
Example:
654
321A
,3211 A 6542 A
,4
11
A ,
5
22
A
6
33A
2005/8 Matrices-6
nmijnmij bBaA ][ ,][
Equal
njmibaBA ijij 1 ,1 ifonly and if
Example:
dc
baBA
43
21
4,3,2,1 ifonly and if dcbaBA
2005/8 Matrices-7
Matrix addition
nmijnmij bBaA ][ ,][
nmijijnmijnmij babaBA ][][][
Example:
31
50
2110
3211
21
31
10
21
2
3
1
2
3
1
22
33
11
0
0
0
2005/8 Matrices-8
Matrix Subtraction
BABA )1(
Scalar Multiplication over a field F (R or C)
F cAA nm ,][
nmijcacA ][
2005/8 Matrices-9
Matrix Multiplication
pnijnmij bBaA ][ ,][
pmijpnijnmij cbaAB ][][][
equalSize of AB
njin
n
kjijikjikij babababac
1
2211
inijii
nnnjn
nj
nj
nnnn
inii
n
cccc
bbb
bbb
bbb
aaa
aaa
aaa
21
1
2221
1111
21
21
11211
2005/8 Matrices-10
05
24
31
A
14
23B
Example:
Sol:
)1)(0()2)(5()4)(0()3)(5(
)1)(2()2)(4()4)(2()3)(4(
)1)(3()2)(1()4)(3()3)(1(
AB
1015
64
19
2005/8 Matrices-11
The partitioned matrices
2221
1211
34333231
24232221
14131211
AA
AA
aaaa
aaaa
aaaa
A
submatrix
3
2
1
34333231
24232221
14131211
r
r
r
aaaa
aaaa
aaaa
A
4321
34333231
24232221
14131211
cccc
aaaa
aaaa
aaaa
A
2005/8 Matrices-12
Properties of Matrix Operations Three elementary matrix operations:
(1) addition
(2) scalar multiplication
(3) multiplication
zero matrix: nm0
identity matrix of order n: nI
2005/8 Matrices-13
The properties of addition and scalar multiplication
(1) A + B = B + A
(2) A + ( B + C ) = ( A + B ) + C
(3) ( cd ) A = c ( dA )
(4) 1A = A
(5) c( A+B ) = cA + cB
(6) ( c+d ) A =cA + dA
thenscalars, are , and ,,, If dcMCBA nm
2005/8 Matrices-14
The properties of zero matrix
Note:(1) 0m×n: the addition identity ( 加法單位矩陣 )
(2) A: the addition inverse ( 加法反元素 ) of matrix A
If AMmn, and c is a scalar, then
(1) A + 0mn = A
(2) A + (A) = 0mn
(3) cA = 0mn c = 0 or A = 0mn
2005/8 Matrices-15
The properties of matrix multiplication
(1) A (BC) = (AB)C
(2) A (B+C) = AB + AC
(3) (A+B)C = AC + BC
(4) c(AB) = (cA) B = A(cB)
The properties of identity matrix
AAI nmMA
n
)1(then
, If
AAI m )2(
2005/8 Matrices-16
The transpose ( 轉置 ) of a matrix
nm
mnmm
n
n
M
aaa
aaa
aaa
A
If
21
22221
11211
mn
mnnn
m
m
T M
aaa
aaa
aaa
A
then
21
22212
12111
2005/8 Matrices-17
Ex: Find the transpose of the following matrices.
8
2A
(b)
987
654
321
A
(c)
11
42
10
A
Sol: (a)
8
2A 82 TA
(b)
987
654
321
A
963
852
741TA
(c)
11
42
10
A
141
120TA
(a)
2005/8 Matrices-18
)4(
)3(
)2(
)1(
TTT
TT
TTT
TT
ABAB
AccA
BABA
AA
The properties of transpose matrices
2005/8 Matrices-19
The symmetric matrix ( 對稱矩陣 )If A = AT , then the square matrix A is called symmetric.
If AT = A , then the square matrix A is called skew-symmetric.
Example:
6
54
321
If
cb
aA is symmetric, then find the values of a, b, c.
Sol:
5 ,3 ,2 cba
The skew-symmetric matrix ( 反對稱矩陣 )
6
54
321
cb
aA
653
42
1
c
ba
AT
TAA
2005/8 Matrices-20
Ex:
0
30
210
If
cb
aA is skew-symmetric, find a, b, c.
Sol:
3 ,2 ,1 cba
Note: TAA is symmetric.
Pf:
symmetric. is
)()(T
TTTTTT
AA
AAAAAA
,
0
30
210
cb
aA
032
01
0
c
ba
AT
TAA
2005/8 Matrices-21
Real Numbers
ab = ba Multiplication commutative
Matrices
BAAB pnnm
undefined. is
defined. is then, If
BA
ABpm (1)
mm
mm
MBA
MABn pm
(3) then, If
nn
nm
MBA
MABnmpm
(2) then, , If
Three possibilities:
2005/8 Matrices-22
Ex: For given matrices A and B, show that AB BA.
12
31A and
20
12B
Sol:
44
52
20
12
12
31AB
BAAB
24
70
12
31
20
12BA
2005/8 Matrices-23
Real numbers
ac = bc, 0c
ba Cancellation laws
Matrices
0 CBCAC
(1) If C is invertible, then A = B
(2) If C is non-invertible, then . (cancellation law does not hold)
BA
2005/8 Matrices-24
Ex: For given matrices A, B and C, show that AC=BC.
21
21 ,
32
42 ,
10
31CBA
Sol:
21
42
21
21
10
31AC
BCAC , but .BA
21
42
21
21
32
42BC