Download Chapter 5 - Set Theory

A very brief introduction to
Matrix
(Section 2.7)
• Definitions
• Some properties
• Basic matrix operations
• Zero-One (Boolean) matrices
Matrix (Section 2.7)
Definition: A matrix is a rectangular array of numbers. A
matrix of m rows and n columns is called an mn matrix,
denoted Amn. The element or entry at the ith row and jth
column is denoted ai,j.
The matrix can also be denoted A = [ai,j].
Example
A3, 2
1 1
 0 2
1 3
column
row
a2,3 = 2
Matrix
Two matrices Amn and Bpq are equal if they have
the same number of rows and columns (m = p and
n = q), and their corresponding entries are equal
(ai,j =bi,j for all i, j).
Amn is a square matrix if m = n, denoted Am
A square matrix A is said to be symmetric if ai,j =
aj,i for all i and j.
1 2 3


A3,3  2 3 4


3 4 1 
Matrix arithmetic (operations)
Matrix addition. Amn and Bmn
• must have the same numbers of rows and
columns
• add corresponding entries
Amn + Bmn = Cmn = [ai,j + bi,j]
A3, 2
5
1 1
 4
 5 6
 0 2 B3, 2   1 6  A3, 2  B3, 2   1 8
1 3
 2  3
 3 0
Matrix subtraction is done similarly
Matrix arithmetic (operations)
Multiply a matrix by a number.
• bA = [bai,j] (i.e., multiply the number to each entry.)
Multiplication of two matrices. Amk and Bkn
• number of columns of the first must equal number of
rows of the second
• the product is a matrix, denoted AB = Cmn
• Entry ci,j is the sum of pair-wise products of the ith row of
A and jth column of B
ci , j  ai ,1b1, j  ai , 2b2, j    ai ,k bk , j
Matrix arithmetic (operations)
Example
A4,3
1 0 4 
14

2
4


2
1
1


8


3 1 0  B3, 2  1 1  AB  C4, 2   7
3 0 
0 2 2
 8
4
9
13
2 
c1,1  a1,1b1,1  a1, 2b2,1  a1,3b3,1  1 2  0 1  4  3  14
c1, 2  a1,1b1, 2  a1, 2b2, 2  a1,3b3, 2  1 4  0 1  4  0  4
c2,1  a2,1b1,1  a2, 2b2,1  a2,3b3,1  2  2  11  1 3  8
Powers and Transposes
Identity matrix: In
• A square matrix of n rows and n columns
• Diagonal entries are 1, all other entries are 0
(ii,i = 1 for all i, ii,j = 0 for all i != j.)
• For matrix Amn, we have Im A = A In = A
1 0 0
I 3  0 1 0 
0 0 1
A3, 2
1 1 
 0 2 
1 3
Powers of (square) matrix An
A0 = In =, Ar = AA···A
r times
I 3  A3, 2
1
 0
1
1
2  A3, 2
3
Powers and Transposes
Matrix transpose: Amn
• the transpose of A, denoted At, is a n m matrix
• At = [bi,j = aj,i]
• ith row of A becomes ith column of At
1 1 
t
1 0 1 
A

A3, 2  0 2
1 2 3
1 3
Theorem: A square matrix An is symmetric iff
A = At
Zero-One (Boolean) Matrix
Definition:
• Entries are Boolean values (0 and 1)
• Operations are also Boolean
Matrix join.
• A  B = [ai,j  bi,j]
Example:
Matrix meet.
• A  B = [ai,j  bi,j]
A  1 0 1 
0 1 0
B  0 1 0 
1 1 0
A  B  1  0 0  1 1  0   1 1 1 
0  1 1  1 0  0 1 1 0
A  B  1  0 0  1 1  0  0 0 0
0  1 1  1 0  0 0 1 0 
Zero-One (Boolean) Matrix
Matrix multiplication: Amk and Bkn
• the product is a Zero-One matrix, denoted AB = Cmn
• cij = (ai1  b1j)  (ai2  b2i)  …  (aik  bkj).
Example:
1
A  0
1
0
1
0
B  1 1 0
0 1 1
1 1
A B  0 1
 1 1
0
1
0