Download Matrix - MSU CSE

Matrices
MSU CSE 260
Outline
• Introduction
• Matrix Arithmetic:
– Sum, Product
• Transposes and Powers of Matrices
– Identity matrix, Transpose, Symmetric matrices
• Zero-one Matrices:
– Join, Meet, Boolean product
• Exercise 2.6
Introduction
Definition A matrix is a rectangular array of
numbers.
 a11 a12
a
a22
21

A

 
a
 m1 am 2
 a1n 
 a2 n 

aij
 
 amn 
element in ith row, jth column
m rows
Also written as A=aij
mn matrix
n columns
When m=n, A is called a square matrix.
Matrix Equality
• Definition Let A and B be two matrices.
A=B if they have the same number of rows
and columns, and every element at each
position in A equals element at
corresponding position in B.
Matrix Addition, Subtraction
Let A = aij , B = bij be mn matrices. Then:
A + B = aij + bij, and A - B = aij - bij
1  1 3 4  4 3
 3 4   1  4    4 0 

 
 

2 0  2 3  4 3
1  1 3 4   2  5
 3 4   1  4    2
8

 
 

2 0  2 3   0  3
Matrix Multiplication
Let A be a mk matrix, and B be a kn matrix,
AB  cij 
k
cij   ait btj  ai1b1 j  ai 2b2 j  ...  aik bkj
t 1
 a11
a
 21
a12
a22
a13 
a23 
b11 b12
b
b
 21 22
b31 b32
b13
b23
b33
b14 
 c11 c12

b24  
 c21 c22
b34 
a11b12  a12b22  a13b32  c12
c13
c23
c14 
c24 
Matching Dimensions
To multiply two matrices, inner numbers must match:
Otherwise,
23 34
not defined.
24 matrix
have to be equal
 a11
a
 21
a12
a22
23
a13 
a23 
b11 b12
b
b
 21 22
b31 b32
b13
b23
b33
34
b14 
 c11 c12 c13

b24  
 c21 c22 c23
b34 
24
c14 
c24 
Multiplicative Properties
Note that just because AB is defined, BA may not be.
Example If A is 34, B is 46, then AB=36, but
BA is not defined (46 . 34).
Even if both AB and BA are defined, they may not have
the same size. Even if they do, matrices do not commute.
1 1
2 1
A
B


2
1
1
1




3 2
 4 3
AB  
BA  


5
3
3
2




however ( AB)C  A( BC ),
assuming dimensions match
Efficiency of Multiplication
 a11
a
 21
23
a12 a13 
a22 a23 
34
b12 b13
b11
b
b22
21

b31 b32
b23
b33
b14 
 c11 c12

b24  
 c21 c22
b34 
c13
c23
c14 
c24 
a11b12 + a12b22 + a13b32 = c12
Takes 3 multiplications, and 2 additions for each element.
This has to be done 24 (=8) times (since product matrix is
24). So 243 multiplications are needed.
•(mk) (kn) matrix product requires m.k.n multiplications.
Best Order?
Let A be a 2030 matrix, B 3040, C 4010. (AB)C or A(BC)?
(2030 3040) 4010
20  40 10  8000 operations
20  30  40  24000 operations
32000
2030 (3040 4010)
30  40 10  12000 operations
20  30 10  6000 operations
So, A(BC) is best in this case.
18000
Identity Matrix
The identity matrix has 1’s down the diagonal, e.g.:
1 0 0
I 3  0 1 0 


0 0 1
1 0 0 a
0 1 0   c


0 0 1  e
For a mn matrix A, Im A = A In
mm mn = mn nn
b  a  0  0 b  0  0 
d   0  c  0 0  d  0 
 

f   0  0  e 0  0  f 
Inverse Matrix
Let A and B be nn matrices.
If AB=BA=In then B is called the inverse of A,
denoted B=A-1.
Not all square matrices are invertible.
Use of Inverse to Solve
Equations
a1 a2
b b
2
 1
 c1 c2
a3   x   p 
b3   y    q 
   
c3   z   r 
a1 x  a2 y  a3 z   p 
 b x  b y  b z   q
2
3
 1
  
 c1 x  c2 y  c3 z   r 
1
1
 x  a 1 a 2
 y   b 11 b 12
   1
1
 z   c 1 c 2
a 13   p 

b 13   q 
 
1
c 3   r 
AX  K
A1 AX  A1 K
IX  A1 K
Please note that a-1j is
NOT necessarily (aj)-1.
Transposes of Matrices





 a ji

aij







Flip across diagonal
1 4 
1 2 3
2 5
written At
 4 5 6




3 6
Transposes are used frequently in various algorithms.
Symmetric Matrix
If At  A
 1 4  1
4 3 0


 1 0 2 
In
A is called symmetric.
is symmetric. Note, for A to be
symmetric, is has to be square.
is trivially symmetric...
Examples
( AB) t  B t At
 k

AB  cij    aix bxj 
 x 1

k


t
( AB) t  cij   c ji    a jx bxi 
 x

k
k
k






B t At   b t ix a t xj    bxi a jx    a jx bxi 
 x 1
  x 1
  x 1

Power Matrix
• For a nn square matrix A, the power matrix
is defined as:
Ar = A  A  …  A
r times
• A0 is defined as In.
Zero-one Matrices
• All entries are 0 or 1.
• Operations are  and .
• Boolean product is defined using:
 for multiplication, and
 for addition.
Boolean Operations
1 0 1
0 1 0 
A
B


0
1
0
1
1
0




0
A  B called "meet" A  B  
0
1
A  B called " join" A  B  
1
0
1
1
1
0
0
1
0
Terminology is from Boolean Algebra.Think
“join” is “put together”, like union, and
“meet” is “where they meet”, or intersect.
Boolean Product
A  aij  be m  k , and B  bij  be k  n
A  B  cij  , cij  (ai1  b1 j )  (ai 2  b2 j )    (aik  bkj )
(Should be a ‘dot’)
1 0
1 1 0


A 0 1 , B



0
1
1


1 0
Since this is “or’d”, you
can stop when you find
a ‘1’
(1  1)  (0  0) 1 0 1 1 0
A  B  (0  1)  (1  0) 1 1  0 1 1

 

(1  1)  (0  0) 1 0 1 1 0
Boolean Product Properties
• In general, A  B  B  A
• Example
1 1 0
0 0 1
A  0 1 1 , B  0 0 1




1 1 0
1 1 1
0 0 1
A  B  1 1 1


0 0 1
1 1 0
B  A  1 1 0


1 1 1
Boolean Power
• A Boolean power matrix can be defined in
exactly the same way as a power matrix.
For a nn square matrix A, the power matrix
is defined as:
A[r] = A  A  …  A
r times
A[0] is defined as In.
Exercise 2.6