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2 - 1
Chapter 2AChapter 2A MatricesMatrices
2A.1 Definition, and Operations of Matrices: 1 Sums and Scalar Products; 2 Matrix Multiplication
2A.2 Properties of Matrix Operations; Mechanics of Matrix Multiplication
2A.3 The Inverse of a Matrix
2A.4 Elementary Matrices
2 - 2
2A.1 Operations with Matrices
Matrix:
11 12 13 1
21 22 23 2
31 32 33 3
1 2 3
[ ]
n
n
nij m n
m m m mn m n
a a a a
a a a a
a a a aA a M
a a a a
L
L
L
M M M M
L
(i, j)-th entry: ija
row: m
column: n
size: m×n
2 - 3
i-th row vector
1 2i i i inr a a a L
j-th column vector
1
2
j
j
j
mj
c
cc
c
M
row matrix
column matrix
Square matrix: m = n
2 - 4
Diagonal matrix:
1 2( , , , )nA diag d d d L
1
2
0 0
0 0
0 0
n n
n
d
dM
d
K
L
M M O M
L
Trace:
If [ ]ij n nA a
11 22Then ( ) nnTr A a a a L
2 - 5
Ex:
654
321A
2
1
r
r
321 ccc
,3211 r 6542 r
,4
11
c ,
5
22
c
6
33c
654
321A
2 - 6
Let [ ] , [ ]ij m n ij m nA a B b
Equal matrix:
if and only if 1 , 1ij ijA B a b i m j n
Ex 1: (Equal matrix)
dc
baBA
43
21
BA If
4 ,3 ,2 ,1 Then dcba
2 - 7
Matrix addition:
If [ ] , [ ]ij m n ij m nA a B b
Then [ ] [ ] [ ]ij m n ij m n ij ij m nA B a b a b
Scalar multiplication:
scalar : ,][ If caA nmij
nmijcacA ][Then
2 - 8
Matrix subtraction:
BABA )1(
Example: Matrix addition
31
50
2110
3211
21
31
10
21
2
3
1
2
3
1
22
33
11
0
0
0
2 - 9
Matrix multiplication:
The i, j entry of the matrix product is the dot product of row i of the left matrix with column j of the right one.
2 - 10
Matrix multiplication:
Notes:Notes: (1) A+B = B+A, (2) BAAB
11 12 111 1 1
21 2 2
1 21 2
11 2
nj n
j n
i i ini i ij in
n nj nnn n nn
a a a b b b
b b ba a a
c c c c
b b ba a a
L L LM M M
M LL
L LM M M MM M M
L LL
2 - 11
Matrix form of a system of linear equations:
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
n n
n n
m m mn n m
a x a x a x b
a x a x a x b
a x a x a x b
L
L
M
L
= = =
A x b
equationslinear m
equationmatrix Single
A x b 1m n n 1m
11 12 1 1 1
21 22 2 2 2
1 2
n
n
m m mn n m
a a a x b
a a a x b
a a a x b
L
L
M M M M M M
L
2 - 12
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
2 - 13
11 12 1
21 22 21 2
1 2
n
nn
m m mn
a a a
a a aA c c c
a a a
L
LL
M M M M
L
1
2
n
x
xx
x
M
11 1 12 2 1
21 1 22 2 2
1 1 2 2 1
n n
n n
m m mn n m
a x a x a x
a x a x a xAx
a x a x a x
L
L
M
L
a linear combination of the column vectors of matrix A:
linear combination of column vectors of A
11 21 1
21 22 21 2
1 2
n
nn
m m mn
a a a
a a ax x x
a a a
LM M M
1c
=
2c
=
nc
=
2 - 14
Keywords in Section 2.1:
row vector: 列向量 column vector: 行向量 diagonal matrix: 對角矩陣 trace: 跡數 equality of matrices: 相等矩陣 matrix addition: 矩陣相加 scalar multiplication: 純量積 matrix multiplication: 矩陣相乘 partitioned matrix: 分割矩陣
2 - 15
2A.2 Properties of Matrix Operations
Three basic matrix operators:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication
Zero matrix:nm0
Identity matrix of order n: nI
2 - 16
Then (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
scalar:, ,,, If dcMCBA nm
Properties of matrix addition and scalar multiplication:
Commutative law for addition
Associative law for addition
Associative law for scalar multiplication
Unit element for scalar multiplication
Distributive law 1 for scalar multiplication
Distributive law 2 for scalar multiplication
2 - 17
calar cMA nm s: , If
AA nm 0 (1)Then
nmA A 0)((2)
nmnm or A c cA 000)3(
Notes:
(1) 0m×n: the additive identity for the set of all m×n matrices
(2) –A: the additive inverse of A
Properties of zero matrices:
2 - 18
Properties of the identity matrix:
AAI
MA
n
nm
(1)Then
If
AAIm (2)
Properties of matrix multiplication:
2 - 19
11 12 1
21 22 2
1 2
If
n
nm n
m m mn
a a a
a a aA M
a a a
L
L
M M M M
L
11 21 1
12 22 2
1 2
Then
m
mTn m
n n mn
a a a
a a aA M
a a a
L
L
M M M M
L
Transpose of a matrix:
AA TT )( )1(TTT BABA )( )2(
)()( )3( TT AccA
)( )4( TTT ABAB
Properties of transposes:
2 - 20
A square matrix A is symmetric if A = AT
Ex:
6
54
321
If
cb
aA is symmetric, find a, b, c?
A square matrix A is skew-symmetric if AT = –A
Skew-symmetric matrix:
Sol:
6
54
321
cb
aA
653
42
1
c
ba
AT
5 ,3 ,2 cba
TAA
Symmetric matrix:
2 - 21
0
30
210
If
cb
aA is a skew-symmetric, find a, b, c?
Note:Note: TAA is symmetric
Pf:
symmetric is
)()(T
TTTTTT
AA
AAAAAA
Sol:
030210
cbaA
032
01
0
c
ba
AT
TAA 3 ,2 ,1 cba
Ex:
2 - 22
ab = ba (Commutative law for multiplication)
undefined. is defined is then , If BAABpm , (1)
mmmm MBAMABnpm , (3) then , If
nnmm MBAMABnmpm , (2) then , , If (Sizes are not the same)
(Sizes are the same, but matrices are not equal)
Real number:
Matrix:
A B B A m n n p
Three situations of the non-commutativity may occur :
Åã¥Ü®à ±.scf
2 - 23
12
31A
20
12B
Sol:
44
52
20
12
12
31AB
24
70
12
31
20
12BA
Example (Case 3):
Show that AB and BA are not equal for the matrices.
and
2 - 24
(Cancellation is not valid)
0 , cbcac
b a (Cancellation law)
Matrix:
0 CBCAC
(1) If C is invertible (i.e., C-1 exists), then A = B
Real number:
BA then ,invertiblenot is C If (2)
2 - 25
21
21 ,
32
42 ,
10
31CBA
Sol:
21
42
21
21
10
31AC
So BCAC
But BA
21
42
21
21
32
42BC
Example: An example in which cancellation is not valid
Show that AC=BCC is noninvertible,
(i.e., row 1 and row 2
are not independent)
2 - 26
Keywords in Section 2.2:
zero matrix: 零矩陣 identity matrix: 單位矩陣 transpose matrix: 轉置矩陣 symmetric matrix: 對稱矩陣 skew-symmetric matrix: 反對稱矩陣
2 - 27
2A.3 The Inverse of a Matrix
nnMA
,such that matrix a exists thereIf nnn IBAABMB
Note:Note:
A matrix that does not have an inverse is called
noninvertible (or singular).
Consider
Then (1) A is invertible (or nonsingular)
(2) B is the inverse of A
Inverse matrix:
2 - 28
If B and C are both inverses of the matrix A, then B = C.
Pf:
CB
CIB
CBCA
CIABC
IAB
)(
)(
Consequently, the inverse of a matrix is unique.the inverse of a matrix is unique. Notes:Notes:(1) The inverse of A is denoted by 1A
IAAAA 11 )2(
TheoremTheorem:: The inverse of a matrix is unique
2 - 29
1nEliminatioJordan -Gauss || AIIA
Example: Find the inverse of the matrix
31
41A
Sol:IAX
10
01
31
41
2221
1211
xx
xx
10
01
33
44
22122111
22122111
xxxx
xxxx
Find the inverse of a matrix AA by Gauss-Jordan Elimination:
2 - 30
1 2 2 1; 41 4 1 1 0 3(1)
1 3 0 0 1 1
M M
M M
1 2 2 1; 41 4 0 1 0 4(2)
1 3 1 0 1 1
M M
M M
1 ,3 2111 xx
1 ,4 2212 xx
11
431AX
Thus
(2) 1304
(1) 0314
2212
2212
2111
2111
xxxx
xxxx
2 - 31
1 2 2 1
Gauss JordanElimination
; 4
1
1 4 1 0 1 0 3 4
1 3 0 1 0 1 1 1
A I I A
M M
M M
If A can’t be row reduced to I, then A is singular.
Note:
2 - 32
326101011
A
1 2
1 1 0 1 0 0
0 1 1 1 1 0
6 2 3 0 0 1
M
M
M
Sol:
1 1 0 1 0 0
1 0 1 0 1 0
6 2 3 0 0 1
A I
M
M M
M
1 36
1 1 0 1 0 0
0 1 1 1 1 0
0 4 3 6 0 1
M
M
M
3
1 1 0 1 0 0
0 1 1 1 1 0
0 0 1 2 4 1
M
M
M
2 34
1 1 0 1 0 0
0 1 1 1 1 0
0 0 1 2 4 1
M
M
M
Example: Find the inverse of the following matrix
2 - 33
3 2
1 1 0 1 0 0
0 1 0 3 3 1
0 0 1 2 4 1
M
M
M
2 1
1 0 0 2 3 1
0 1 0 3 3 1
0 0 1 1 4 1
M
M
M
So the matrix A is invertible, and its inverse is
1 [ ]I A M
You can check: IAAAA 11
1
2 3 1
3 3 1
1 4 1
A
2 - 34
0(1) A I
factors
(2) ( 0)k
k
A AA A k L144424443
(3) , : integersr s r sA A A r s
( )r s rsA A
1 1
2 2
0 0 0 0
0 0 0 0(4)
0 0 0 0
k
kk
kn n
d d
d dD D
d d
L LL L
M M O M M M ML L
Power of a square matrixsquare matrix:
2 - 35
If A is an invertible matrix, k is a positive integer, and c is a scalar,
then 1 1 1(1) is invertible and ( )A A A
1 1* 1 1 1 1
factors
(2) is invertible and
( ) ( )
k
k k k k
k
A
A A A A A A A L1444442444443
1 11(3) c is invertible and ( ) , 0A cA A c
c
1 1(4) is invertible and ( ) ( )T T TA A A
TheoremTheorem :: Properties of inverse matrices
2 - 36
TheoremTheorem:: The inverse of a product
If A and B are invertible matrices of size n, then AB is invertible and
111)( ABAB
111)( So
unique. is inverse its then ,invertible is If ABAB
AB
Pf:
IBBIBBBIBBAABABAB
IAAAAIAIAABBAABAB
1111111
1111111
)()()())((
)()()())((
Note:Note:
1 1 1 1 11 2 3 3 2 1n nA A A A A A A A
L L
2 - 37
Theorem:Theorem: Cancellation properties
If C is an invertible matrix, then the following properties
hold:
(1) If AC=BC, then A=B (Right cancellation property)
(2) If CA=CB, then A=B (Left cancellation property)
Pf:
BA
BIAI
CCBCCA
CBCCAC
BCAC
)()(
)()(11
11 exists) C so ,invertible is (C 1-
Note:If C is not invertible, then cancellation is not valid.
2 - 38
TheoremTheorem:: Systems of linear equations with unique solutions
If A is an invertible matrix, then the system of linear equations
Ax = b has a unique solution given by1x A b
Pf:
( A is nonsingular)
bAx
bAIx
bAAxA
bAx
1
1
11
This solution is unique.
.equation of solutions twowere and If 21 bAxxx
21then AxbAx 21 xx (Left cancellation property)
2 - 39
Keywords in Section 2.3:
inverse matrix: 反矩陣 invertible: 可逆 nonsingular: 非奇異 singular: 奇異 power: 冪次
2 - 40
Note:Note:
Only do a single elementary row operation.
2A.4 Elementary Matrices
Row elementary matrix:
An nn matrix is called an elementary matrixelementary matrix if it can be obtained
from the identity matrix I by a single elementary row operation.
2 - 41
100030001
)(a
010001)(b
000010001
)(c
010100001
)(d
1201)(e
100020001
)( f
Yes 2 3(3 ( ))I square)(not Not)constan nonzero aby bemust
tionmultiplica (Row No
Y 2 3 3es ([ ]( ))I Y 1 2 2es ([2 ]( ))I
)operations row
elementary two(Use No
Ex: (Elementary matrices and nonelementary matrices)
2 - 42
Notes:Notes:
(1) ( )ij ijA P A
(2) k ( ) ( )i iA M k A
i(3) [k ]( ) ( )j ijA C k A
( )
( )
I E
A EA
Theorem:Theorem: Representing elementary row operations
Let E be the elementary matrix obtained by performing an
elementary row operation on Im. If that same elementary row
operation is performed on an mn matrix A, then the
resulting
matrix is given by the product EA.
2 - 43
0262
2031
5310
A
Sol:
1 12 3
0 1 0
( ) 1 0 0
0 0 1
E I
2 1 3 3
1 0 0
[ 2 ]( ) 0 1 0
2 0 1
E I
3 3 3
12
1 0 01
[ ]( ) 0 1 02
0 0
E I
Example: Using elementary matrices
Find a sequence of elementary matrices that can be used to write
the matrix A in row-echelon form.
2 - 44
1 12 1
0 1 0 0 1 3 5 1 3 0 2
( ) 1 0 0 1 3 0 2 0 1 3 5
0 0 1 2 6 2 0 2 6 2 0
A A E A
2 1 3 1 2 1
1 0 0 1 3 0 2 1 3 0 2
[ 2 ]( ) 0 1 0 0 1 3 5 0 1 3 5
2 0 1 2 6 2 0 0 0 2 4
A A E A
3 3 2 3 2
1 0 0 1 3 0 2 1 3 0 21
[ ]( ) 0 1 0 0 1 3 5 0 1 3 52
1 0 0 2 4 0 0 1 20 0
2
A A E A
3 2 1 B E E E A 3 1 3 12
1or [ ][ 2 ] ( )
2B A
B
row-echelon form
2 - 45
Matrix B is row-equivalent to A if there exists a finite number
of elementary matrices such that
1 2 1k kB E E E E A L
Row-equivalent:Row-equivalent:
2 - 46
TheoremTheorem:: Elementary matrices are invertible
If E is an elementary matrix, then exists and
is an elementary matrix.
Notes:Notes:1(1) (P )ij ijP
1 1(2) [ ( )] ( )i iM k M
k
1(3) [C ( )] ( )ij ijk C k
1E
2 - 47
1 12 12
0 1 0
( ) 1 0 0
0 0 1
E P
1 112 1
0 1 0
( ) 1 0 0
0 0 1
P E
2 1 3 13
1 0 0
( 2 ) 0 1 0 ( 2)
2 0 1
E C
1 113 2
1 0 0
[ ( 2)] 0 1 0
2 0 1
C E
3 3 3
12
1 0 01 1
( ) 0 1 0 ( )2 2
0 0
E M
1 13 3
1 0 01
[ ( )] 0 1 02
0 0 2
M E
12P
13(2)C
3(2)M
Ex:
Elementary Matrix Inverse Matrix
2 - 48
Pf:Pf: (1) Assume that A is the product of elementary matrices.
(a) Every elementary matrix is invertible.
(b) The product of invertible matrices is invertible.
Thus A is invertible.
(2) If A is invertible, has only the trivial solution.0xA 0 0A I M M
3 2 1 kE E E E A I L1 1 1 1
1 2 3 kA E E E E L
Thus A can be written as the product of elementary matrices.
Theorem:Theorem: A property of invertible matrices
A square matrix A is invertible if and only if it can be written as
the product of elementary matrices.
2 - 49
83
21A
Sol:
1 1 2
22 1
3
1[ 2 ]2
1 2 1 2 1 2
3 8 3 8 0 2
1 2 1 0
0 1 0 1
A
I
21 2 12 1
1Therefore ( 2) ( ) ( 3) ( 1)
2C M C M A I
Ex:
Find a sequence of elementary matrices whose product is
2 - 50
1 1 1 11 12 2 21
1Thus [ ( 1)] [ ( 3)] [ ( )] [ ( 2)]
2A M C M C
1 12 2 21 ( 1) (3) (2) (2)M C M C
10
21
20
01
13
01
10
01
Note:Note:If A is invertible
13 2 1[ ] [ ]kE E E E A I I A L M M
3 2 1Then kE E E E A IL
13 2 1kA E E E E L
2 - 51
If A is an nn matrix, then the following statements are equivalent.
(1) A is invertible.
(2) Ax = b has a unique solution for every n1 column matrix b.
(3) Ax = 0 has only the trivial solution.
(4) A is row-equivalent to In .
(5) A can be written as the product of elementary matrices.
Thm:Thm: Equivalent conditions
2 - 52
LUAL is a lower triangular matrix
U is an upper triangular matrix
A nn matrix A can be written as the product of a lowerlower
triangular matrixtriangular matrix LL and an upper triangular matrixupper triangular matrix UU, such as
Note:Note:
If a square matrix A can be row reduced to an upper triangular
matrix U using only the row operation of adding a multiple of one only the row operation of adding a multiple of one
rowrow to another (pivot)to another (pivot), then it is easy to find an LU-factorization of A.
2 1
1 1 11 2
k
k
E E E A U
A E E E U
A LU
L
L
LLUU-factorization:-factorization:
2 - 53
01
21 )( Aa
2102310031
)( Ab
1 211 2 1 2
1 0 0 2A U
Sol: (a)
12( 1)C A U
112 12[ ( 1)] (1)A C U C U LU
112 12
1 0[ ( 1)] (1)
1 1L C C
Ex: Ex: LULU-factorization-factorization
2 - 54
(b)
1 3 2 32 4
1 3 0 1 3 0 1 3 0
0 1 3 0 1 3 0 1 3
2 10 2 0 4 2 0 0 14
A U
23 13(4) ( 2)C C A U
1 113 23[ ( 2)] [ (4)]A C C U LU
1 113 23 13 23[ ( 2)] [ (4)] (2) ( 4)
1 0 0 1 0 0 1 0 0
0 1 0 0 1 0 0 1 0
2 0 1 0 4 1 2 4 1
L C C C C
2 - 55
If thenA LU LUx b
Let theny Ux Ly b
Two steps:Two steps:
(1) Write y = Ux and solve Ly = b for y
(2) Solve Ux = y for x
bAx
Solving Ax=bAx=b with an LU-factorization of A
2 - 56
Sol:
LUA
1400
310
031
142
010
001
2102
310
031
2021021353
321
32
21
xxxxx
xx
bLyUxy solve and,Let )1(
2015
142010001
3
2
1
yyy
14)1(4)5(220
422015
213
2
1
yyy
yy
Ex: Solving a linear system using LU-factorization
2 - 57
yUx system following theSolve )2(
1415
1400
310031
3
2
1
xxx
1)2(3535
2)1)(3(131
1
21
32
3
xx
xx
x
Thus, the solution is
1
2
1
x
So
2 - 58
Keywords in Section 2A.4:
row elementary matrix: 列基本矩陣 row equivalent: 列等價 lower triangular matrix: 下三角矩陣 upper triangular matrix: 上三角矩陣 LU-factorization: LU 分解