Download Introduction to Matrices

MATH 1046
Introduction to Matrices
(Sections 3.1 and 3.2)
Alex Karassev
Matrices
• An m x n matrix is a rectangular array of
numbers with m rows and n columns
j-th column
 a11

 a21
A   ai1

 am11
 a
 m1
a12
a22
...
...
a1 j
a2 j
...
...
ai 2
...
aij
...
am11
...
am1 j
...
am 2
...
am j
...
a1n 

a2 n 
ain  i-th row

am1 n 
am n 
Some examples
•
•
•
•
•
•
Coefficient matrix of a linear system
Distance matrix
Graph adjacency matrix
Matrices in computer graphics
Matrices in optimization problems
Games
Example: distance matrix (in km)
North Bay
Toronto
Ottawa
North Bay
0
340
360
Toronto
340
0
450
Ottawa
360
450
0
Example: graph adjacency matrix
1

1
0

0

Timmins
1
North Bay 2
Ottawa
3
Toronto
4
1
1
1
1
0
1
1
1
0

1
1

1 
Matrix operations
•
•
•
•
Multiplication by scalars
Addition and subtraction
Multiplication
Transpose
Multiplication by scalars
• Example: distance matrices in km and mi
km
NB
T
O
mi
NB
T
O
NB
0
340
360
NB
0
340/1.6
360/1.6
T
340
0
450
T
340/1.6
0
450/1.6
O
360
450
0
O
360/1.6
450/1.6
0
• All entries are multiplied by 1/1.6
Multiplication by scalars
 a11

 a21
A   ai1

 am 11
 a
 m1
a12
a22
ai 2
am 11
am 2
 ca11

 ca21
cA   cai1

 cam 11
 ca
 m1
ca12
ca22
cai 2
cam 11
cam 2
...
...
...
...
...
...
...
...
...
...
a1 j
a2 j
aij
am 1 j
am j
ca1 j
ca2 j
caij
cam 1 j
cam j
a1n 

a2 n 
ain 

am 1 n 
am n 
...
...
...
...
...
...
...
...
...
...
ca1n 

ca2 n 
cain 

cam 1 n 
cam n 
Addition
First
Apples
Basket
Second
Apples
Basket
Pears
Pears
Red
3
5
Red
1
0
Green
4
6
Green
10
3
Total
Apples
Pears
Red
3+1
5+0
Green
4+10
6+3
Addition
 a11
a
A   21
 ...
a
 m1
a12
a22
...
am 2
... a1n 
... a2 n 
... ... 
... amn 
 a11  b11
 a b
21
21

A B 
 ...
a  b
 m1 m1
 b11 b12 ... b1n 
b

...
b
b
2n 
22
B   21
 ... ... ... ... 
b

b
...
b
mn 
 m1 m 2
a12  b12
a22  b22
...
am 2  bm 2
... a1n  b1n 
... a2 n  b2 n 

... ...
... amn  bmn 
Matrix miultiplication
• Example: 2 x 2 linear system
a11 x1  a12 x 2 c1

a21 x1  a22 x 2 c 2
 a11 a12 


 a21 a22 
• Linear substitution
 x1  b11 y1  b12 y 2

 x2  b21 y1  b22 y 2
 b11 b12 


 b21 b22 
 x1  b11 y1  b12 y 2

 x2  b21 y1  b22 y 2
a11x1a12 x 2 c1

a21x1a22 x 2 c 2
 b11 b12 


 b21 b22 
 a11 a12 


 a21 a22 
Substitution:
a11 (b11 y1b12 y 2 )  a12 (b21 y1  b22 y 2 ) c1

a21 (b11 y1b12 y 2 )  a22 (b21 y1  b22 y 2 ) c 2
What system do we get in terms of y1 and y2?
a11 (b11 y1b12 y 2 )  a12 (b21 y1  b22 y 2 ) c1

a21 (b11 y1b12 y 2 )  a22 (b21 y1  b22 y 2 ) c 2
(a11b11  a12b21 ) y1(a11b12  a12b22 ) y 2 c1

(a21b11  a22b21 ) y1(a21b12  a22b22 ) y 2 c 2
New Coefficient Matrix:
 a11b11  a12b21

 a21b11  a22b21
a11b12  a12b22 

a21b12  a22b22 
Matrix multiplication: 2 x 2 case
 a11
A  
 a21
a12 

a22 
 a11b11  a12b21
AB  
 a21b11  a22b21
 b11 b12 

B  
 b21 b22 
a11b12  a12b22 

a21b12  a22b22 
Matrix multiplication: 2 x 2 case
 a11
A  
 a21
a12 

a22 
Dot product:
 a11b11  a12b21
AB  
 a21b11  a22b21
 b11 b12 

B  
 b21 b22 
a11b12  a12b22 

a21b12  a22b22 
Matrix multiplication: general case
• Let A = (aij) be m x n matrix and B be n x k
matrix
• The product AB = C = (cij) is an m x k matrix
defined as follows
cij = ai1 b1j + ai2 b2j+ ci3 b3j+…+ain bnj
• Note: cij is the dot product of i-th row of A and
j-th column of B
Is it possible that AB≠BA?
Is it possible that AB≠BA?
Yes!
0 1

A  
1 0
 1 1

B  
 0 1
 0 1

AB  
 1 1
1 1 

BA  
1 0 
Square matrix
• A matrix is called square if m=n, i.e. the number
of rows is the same as the number of columns
• A square matrix has the diagonal: all entries of
the form aii
 a11

 a21
a
 31
a12
a22
a32
a13 

a23 

a33 
Zero Matrix and Identity Matrix
• Zero matrix: an n x n matrix such that all
entries are zeros
• Identity matrix: an n x n matrix such that all
diagonal entries are 1, all other entries are 0
0 0 0


O  0 0 0
0 0 0


1 0 0


I  0 1 0
0 0 1


Properties: addition and multiplication by scalars
• Addition of matrices and multiplication by scalars have the same
properties as in the case of vectors or real numbers:
• (A+B)+C=A+(B+C)
• A+B = B+A
• A+O = A
• If -A = (-1)+A then A+(-A) = O
• (cd)A= c(dA)
• 1A = A
• (c+d)A = cA + dA
• c(A+B) = cA +cB
Properties: matrix multiplication
• Assuming the dimension of matrices allow to
perform the operations, we have the following:
•
•
•
•
•
•
(AB)C=A(BC)
A(B+C)= AB+AC
(B+C)A= BA+CA
IA = A I = A
OA = AO = O
(cA)B= A (cB) = c(AB)
Powers
• For a square matrix A, define Ak = A A … A
(product of A with itself k times)
• Question: Is it possible to have A2 = O for a
non-zero A?
Yes!
0 1

A  
0 0
Scalar matrices
• Matrix A is called scalar if A = aI for some real
number a
• Exercise
a) Prove that for a scalar n x n matrix A and for any
n x n matrix B we have AB= BA = aB
b) Prove (at least for 2 x 2 case) that in fact for an
arbitrary square matrix A we have AB= BA for any
B if and only if A is scalar
Matrix Transpose
• AT = B such that bij = aji
• If A is m x n matrix AT is n x m matrix
• If v is a row vector vT is a column vector, and vice versa
• In general:
columns of AT are rows of A
rows of AT are columns of A
• For a square matrix A this can be viewed as a flipping with
respect to the diagonal:
 a11

A   a21
a
 31
a12
a22
a32
a13 

a23 
a33 
 a11

T
A   a12
a
 13
a21
a22
a23
a31 

a32 
a33 
Properties of transpose operation
•
•
•
•
•
(AT)T = ?
(cA) T= ?
(A+B)T = ?
(AB)T = ?
(Ak)T= ?
Properties of transpose operation
•
•
•
•
•
(AT)T = A
(cA) T=cAT
(A+B)T = AT+ BT
(AB)T =BTAT
(Ak)T= (AT) k
• Note: in general AAT ≠ATA (exercise: find an
example!); square matrices that commute
with its transpose are called normal
Symmetric and skew-symmetric
matrices
• A square matrix is called
 Symmetric if AT = A
• Examples: distance matrix, adjacency matrix
 Skew-symmetric (or antisymmetric) if AT = -A
• Examples: matrices of some games
Symmetric and Skew-symmetric matrices
• (A+AT)T= AT + A = A + AT
• (AAT)T=(AT)TAT =AAT
A+AT and AAT are symmetric for any square A
• (A-AT)T=AT - A = -(A - AT)
A-AT is skew-symmetric for any square A
• Note: for any square matrix A we have
A=1/2(A+AT) + 1/2 (A-AT)
Symmetric matrices: Exercise
• Give an example of two symmetric 2x2
matrices whose product is not symmetric
• Prove that the product of two symmetric
square matrices A and B is symmetric if and
only if AB = BA