Date post: | 01-Nov-2014 |
Category: |
Documents |
Upload: | osamahussien |
View: | 11 times |
Download: | 4 times |
A matrix is an ordered rectangular array of numbers. The size of a matrix is given by the numbers. The size of a matrix is given by the number of rows and the number of columns.Let m denote the number of rowsLet m denote the number of rowsLet n denote the number of columns.Let aij denote the entry in the ith row and the jth column.jth column.
nm
naaa
nm
11211
n
n
aaa
aaa
22221
11211
naaa 22221m rows
mnmm aaa 21 mnmm aaa 21
n columns
Zero matrixijij
square matrix square matrix
The Identity Matrix, denoted , is a diagonal matrix of order nxn with all the
nI
diagonal matrix of order nxn with all the diagonal entries equal to 1.
00
00
000
000
22
11
d
d
DO
00
00
000
000 22
d
dDO
mm
001
00000 dmm
matrixidentity an denote Let 010
001
nnII n
100
n
A square matrix all its elements below the main diagonal are zeros. 0
i ja
0
ijai j
2 3 3
0
0 0
7 1
9
A
0 0 9
A square matrix all its elements above the main diagonal are zeros. 0
i ja
0
ijai j
2 0 0
5 7
2 8 9
0A
2 8 9
The matrix A is 3x3 of the form
2
i>ji2
i>j
i jij
ia
i j
i ji j
2 3 411 12 13a a a
4 4 5
6 6 6
A11 12 13
21 22 23
a a a
A a a a
a a a6 6 6
31 32 33a a a
2
5
i i j
a i j
5ija i j
i j i j
3
i j i j
j i j
3ij
j i jb
i j i j
i j i j
If
A and B are 3x3 find A-2B
5 1 2a a a1 1 1 2 1 3
2 1 2 2 2 3
5 1 2
4 5 1
a a a
A a a a
3 1 3 2 3 3 6 6 5
3 3 4
a a a
3 3 4
3 6 5
3 6 9
B
3 6 9
1 7 1 0
2 2 7 1 1A B2 2 7 1 1
0 6 1 3
A B
Equality of Matrices:Two matrices are equal if they have the Two matrices are equal if they have the
same size and their corresponding entries are equal. are equal.
Find x and y that satisfies the following equationequation
x y y yx y y y
y
y=2y=2x+6=2 x=-4
A new matrix C may be defined as the additive combination of matrices A and additive combination of matrices A and B where: C = A + Bis defined by:is defined by:
ij ij ij
1,2,..., 1,2,...,and
ij ij ij
i m j n1,2,..., 1,2,...,and
i m j n
Note: Only matrices of the same dimension can Note: Only matrices of the same dimension can be added
Addition of Matrices
nn bbbaaa 1121111211
n
n
n
n
bbb
bbb
Baaa
aaa
A 22221
11211
22221
11211
and
mnmmmnmm bbbaaa 2121
nn
bababa
bababa 1112121111
nn
bababa
bababaBA 2222222121
mnmnmmmm bababa 2211
A
B
3
4
1
2
4
6
CA
B
5 6
3 4
8 10
C
Multiplying a matrix by a real number (scalar) results Multiplying a matrix by a real number (scalar) results in a matrix with each entry multiplied by the scalar. Let k be a real number.
naaa 11211 nkakaka 11211
k be a real number.
n
n
aaa
aaa
A 22221
11211
n
n
kakaka
kakaka
kA 22221
11211
mnmm aaa 21 mnmm kakaka 21
k(A + B) = kA + k B(x+y) A = x A + y A(x+y) A = x A + y A(x y) A = x ( y A)o A = Ox O = Ox O = O
C = A - BC = A - Bis defined byC = A + (-1) BC = A + (-1) B
nn bababa 1112121111
nn
nn
bababa
bababa
BA 2222222121
1112121111
bababa
BA
mnmnmmmm bababa 2211
The scalar matrix can be written as .
C = InC = In
0 0 0x 0 0 0
0 0 0
x
x0 0 0
0 0 0
x
x0 0 0
0 0 0
x
x
Matrices A and B have these dimensions:
Matrices A and B can be multiplied if:
The resulting matrix will have the dimensions:
pn
bbb
bbb
aaa
aaa 1121111211
pn
bbb
bbbB
aaa
aaaA
2222122221 and
npnnmnmm bbbaaa 2121
ccc
22221
11211
p
p
ccc
ccc
CAB
Entry cij is obtained by taking the sum of the products of the entries of
21 mpmm ccc
CABproducts of the entries of the ith row in A with the jth column in B.
122122112121
112112111111
...
...Entry
nn
nn
bababac
bababacjth column in B.
212212121122
122122112121
...
...
nn
nn
bababac
bababac
mxn mxs sxnmxn mxs sxn
s
1
s
ij ik kjk
c a b1
1, 2,...,k
i m
1, 2,...,j n
A
2
3
1
1
and
B
1 1 1 A
1
1
1
0
and
B
1 0 21
0[3 x 2] [2 x 3]
A and B can be multiplied
[3
x 3][3
x 3]
2*1 3*0 2
5 2 82*1 3*1 5 2*1 3*2 8
2*1 3*0 2
5 2 8
1*1 1*1 2 1*1 1*2 3 2 1 3
2*1 3*1 5 2*1 3*2 8
1*1 1*0 1C
1*1 0*0 1 1 11*1 0*1 1 1* 0 11 *2 1
2x3 3x2 2x2
2 31 1 1
1 1
B x A 1 11 0 2
1 0
B x A
4 44 4
4 3 =
Properties of Matrix Multiplication
1.AB BA
AB C A BC
2-
3. )
AB C A BC
A B C AB AC
2-
(
4. A m n I I
(B+C)A=BA+CA
If is an matrix and and are4. m nA m n I I
m m n n
If is an matrix and and are
and identity matrices, respectively,
I A AI A
then
m nI A AI A
5- Amxn 0nxk =0mxk
0kxm Amxn = 0kxn
If A is an m x n matrix with elements aij, then the transpose of A, denoted, AT, is an n x m matrix with elements aji.
aaa aaa
n
n
aaa
aaa
A 22221
11211
m
m
T aaa
aaa
A 22212
12111
mnmm aaa
A
21 mnnn
T
aaa
A
21mnmm aaa 21 mnnn aaa 21
TT TT
T T T
T T T
T T
A square matrix B is said to be symmetric if
B = BTB = BT
brs = bsr s rr=1,2, ,ms=1,2, ,ns=1,2, ,n
Any diagonal matrix is symmetric
A squared zero matrix is symmetric
The transpose of an upper triangular matrix is The transpose of an upper triangular matrix is a lower triangular matrix
A squared matrix is said to skew-symmetric ifA = - AT
0 1 2
1 0 3A 1 0 3
2 3 0
A
0 1 2
1 0 3TA 1 0 3
2 3 0
TA
TA A
The common term of a skew-symmetric matrix can be written as
r sars
sr
r sa
a r s
If the common element of a 3x3 matrix is given by
rs
Show that A is a skew-symmetric matrixShow that A is a skew-symmetric matrix
0 1 10 1 1
1 0 1A 1 0 1
1 1 0
A
0 1 10 1 1
1 0 1TA
1 1 0TA A TA A
X =(x1,x2, .,xn) 1xnAT =(1,1, .,1) 1xnAT =(1,1, .,1) 1xn
xi = X A = AT XT
xi2 = X XTxi = X X
A square matrix is said to be orthogonal ifT TT T
T H E I D E N T I T Y M AT R I X I S O R T H O G O N AL AN D S YM M E T R I CO R T H O G O N AL AN D S YM M E T R I C
- 1 - 1- 1 - 1
2 2A = 2 2A =1 - 1
2 22 2
- 1 - 11 - 1 - 11A =
1 - 12
T - 1 11A =
- 1 - 12 - 1 - 12
T TT T2
If A is a square matrix and k is a positive integer, the power k of A is defined asinteger, the power k of A is defined asAk = A . A ..A
So, A2 = A . ASo, A = A . Aand A0 = I
A square matrix A is said to be idempotent if
Ak = A for any positive integer k
The identity matrix is idempotentThe identity matrix is idempotentThe square zero matrix is idempotent
Show that the matrix A defined below is idempotent 1 1
2 21 1
A1 1
2 2
1 1 1 1 1 1
A
2
1 1 1 1 1 1
2 2 2 2 2 21 1 1 1 1 1
A A A A1 1 1 1 1 1
2 2 2 2 2 2
3 2A A A A A A
Assume Ak-1 = AAk = Ak-1 A =A A=AAk = Ak-1 A =A A=ASo Ak = A for any positive integer ki.e. A is idempotent
Definition: Let A be an n n matrix. An inverse of A Definition: Let A be an n n matrix. An inverse of A is an n n matrix B such that:
AB = In and BA = InAB = In and BA = In
A is then called invertible.A is then called invertible.B is called an inverse of AIf A has no inverse, it is called singular.If A has no inverse, it is called singular..
The inverse (if exists) is unique.The inverse (if exists) is unique.Proof:Assume A is invertible, with two inverses B and C, i.e.Assume A is invertible, with two inverses B and C, i.e.A B = B A = IandandAC = CA = I,
(BA)C=B(AC)=BI=B(BA)C=IC=C(BA)C=IC=C
Thus, B=C Thus, B=C
Since the inverse ,if it exists, is unique, we call it A-1call it A
A A-1 = A-1 A = IA A-1 = A-1 A = In
Let A and B be invertible matrices of the same size, and k be a nonzero scalar. Then:and k be a nonzero scalar. Then:
1. I 1 = I
(A 1) 1 = A2. (A 1) 1 = A
3. (kA) 1 = k 1A 1
4- (AB) 1 = B 1A 1
5- (An) 1 = (A 1)n5- (A ) = (A )
(AT) 1 = (A 1)T6- (AT) 1 = (A 1)T
7- If A is orthogonal then it is invertible andA-1=ATA =A
If A and B are nxn invertible matrices, prove that AB is invertible also.that AB is invertible also.
(AB)(AB)-1=(AB)(B-1A-1) = A(BB-1)A-1(AB)(AB)-1=(AB)(B-1A-1) = A(BB-1)A-1
n-1 -1
n(AB)-1 (AB) = (B-1A-1) (AB)= B(AA-1)B-1(AB)-1 (AB) = (B-1A-1) (AB)= B(AA-1)B-1
n-1 -1
nn n
Find the inverse of the 2 2 matrix:
1
1
1
1
If3 1
5 2A
Show that
5 2
Show that A2 -5A+I2=0Using this result find A-1Using this result find A-1
A2 - 5A + I2 =0A-1 A A 5
A-1 A + A-1 = 0A-1 A A 5
A-1 A + A-1 = 0A 5 I + A-1 = 0A-1 = 5 I A
-1
3 53 5
2 3 A =If
4 71 4 7
2 4B
a m atrix X su ch th atFind
a m atrix X su ch th at
X B =A
Find
X B = AX B B-1 = A B-1X B B-1 = A B-1
X I = A B-1
Let A, B, and X be 3
invertible matrices of Let A, B, and X be 3
invertible matrices of the same size. Solve the following matrix the same size. Solve the following matrix equation for X:
(A 1XB) 1
= (BA)2(A 1XB) 1
= (BA)2
Note: Be careful with the order of the matrix Note: Be careful with the order of the matrix multiplication.
Answer: X = (B2AB) 1
= B 1A 1(B 1)2Answer: X = (B2AB) 1
= B 1A 1(B 1)2
(A-1 X B)-1 = (B A)2
B-1 X-1 (A-1 )-1 = (B A)2B-1 X-1 (A-1 )-1 = (B A)2
B-1 X-1 A = (BA)2
B B-1 X-1 A = B (BA)2
I X-1 A A-1 = B (BA)2 A-1I X A A = B (BA) AX-1 I = B BA BA A-1
X-1 = B2 A BX-1 = B2 A BX = ( B2 A B)-1X = ( B A B)
Show that the inverse of the general 2 2
matrix:Show that the inverse of the general 2 2
matrix:
baA
dc
baA
dc
bdA 1
ac
bd
bcadA 1
1
1
- 1 x yA s s u m e A =
w z
- 1
A s s u m e A =w z
a b x y 1 0A A = =
c d w z 0 1A A = =
c d w z 0 1
a x + b w = 1
c x + d w = 0c x + d w = 0
- c a x - c b w = - c
c a x + a d w = 0c a x + a d w = 0
- cw = a d - b c 0
a d - b cd
x = a d - b c 0a d b c