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Chapter 3
3.8 Matrices
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Matrix Arithmetic
Algorithms for Matrix Multiplication
Transposes and Powers of Matrices
Zero-One Matrices
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Matrix Arithmetic
• Definition 1:
• A matrix is a rectangular array of numbers.
• A matrix with m rows and n columns is called an
m × n matrix.
• The plural of matrix is matrices. A matrix with the
same number of rows as columns is called square.
• Two matrices are equal if they have the same
number of rows and the same number of columns
and the corresponding entries in every position are
equal.
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Matrix Arithmetic
• Definition 2: Let
 a11
a
A   21
 

 an1
a12
a22
• The ith row of A is the
1 x n matrix [ai1,ai2,. . .,ain].
 a1n  • The jth column of A is
 a2 n 
the n x 1 matrix  a1 j 
a 
 2j
an 2 
  



The (i, j)th element or entry of is the element aij ,
 a nj 

that is , the number in the ith row and jth column of A.
A convenient shorthand notation for expressing the
matrix A is to write A =[aij], which indicates that A is the
matrix with its (i, j)th element equal to aij.


 

ann 
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Matrix Arithmetic
• Definition 3: Let A=[aij] and B=[bij] be m x n
matrices. The sum of A and B, denoted by A+B,
is the m x n matrix that has aij+bij as its (i, j)th
element. In other words, A+B= [aij+bij].
• Example 2: we have
1
2

3
0
2
4
 1  3
 3   1
0   1
4
3
1
4
 1


0 3

2 
2
 2

 1  3
5
2 
4
4
Matrix Arithmetic
• Definition 4:
• Let A be an m x k matrix and B be k x n matrix.
• The product of A and B, denoted by AB, is the m x n
matrix with its (i , j )th entry equal to the sum of the
products of the corresponding elements from the
ith row of A and the jth column of B.
• In other words, if AB=[cij], then
cij = ai1b1j + ai2b2j +. . . +aikbkj
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Matrix Arithmetic
6
Algorithms for Matrix Multiplication
• Algorithm 1 : Matrix
Multiplication
procedure matrix multiplication
(A, B: matrices)
for i := 1 to m
for j := 1 to n
begin
cij :=0
for q := 1 to k
cij :=cij + aiqbqj
end
{C= [cij] is the product of A and B}
• Example 6: In which order
should the matrices A1, A2,
and A3, where
• A1 is 30x20 , A2 is 20x40 ,
A3 is 40x10,
• all with integer entries –
be multiplied to use the
least number of
multiplications of integers?
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Transposes and Powers of Matrices
• Definition 5:
the identity matrix of order n is the n x n matrix
In = [δij]
where δij =1 if i = j and δij = 0 if i ≠ j. Hence,
1 0 0 0
0 1 0 0 

In  
   


0 0 0 1 
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Transposes and Powers of Matrices
• Definition 6: Let A=[aij] be an m x n matrix.
• The transpose of A, denoted by At, is the n x m
matrix obtained by interchanging the rows and
columns of A .
• In other words, if At=[bij], then bij = aji for i=1,2,. . .,n
and j = 1,2,. . .,m .
• Definition 7: A square matrix A is called symmetric
if A = At.
• Thus A =[aij] is symmetric if aij = aji for all i and j with
1≤ i ≤ n and 1 ≤ j ≤ n .
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Symmetric Matrix
10
Zero-One Matrices
• Definition 8: Let A=[aij] and B=[bij] be m x n zeroone matrices.
• Then the join of A and B is the zero-one matrix with
(i , j )th entry aij v bij.
The join of A and B is denoted by A v B.
• The meet of A and B is the zero-one matrix with (i ,
j )th entry aij Λ bij.
The meet of A and B is denoted by A Λ B.
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Zero-One Matrices
• Definition 9: Let A=[aij] be an m x k zero-one matrix
and B=[bij] be a k x n zero-one matrix .
• Then the boolean product of A and B,denote by A⊙B ,
is the m x n matrix with with (i , j)th entry cij where
cij = (a i1  b1j )  ( a i2  b 2j )  . . .  (a ik  b kj )
• Example 10: find the Boolean product of A and B,
1 0
where
1 1 0


A  0 1, B  

0
1
1


1 0
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Zero-One Matrices
• Algorithm 2: The Boolean Product
procedure Boolean product(A, B: zero-one matrices)
for i := 1 to m
for j := 1 to n
begin
cij :=0
for q := 1 to k
cij := cij  (a iq  b qj )
end
{C= [cij] is the Boolean product of A and B}
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Zero-One Matrices
• Definition 10: Let A be a square zero-one
matrix ant let r be a positive integer.
• The rth Boolean power of A is the Boolean
product of r factors of A. The rth Boolean
product of A is denoted by A[r]
• Hence, A[r]  A ⊙ A ⊙ A ⊙⊙ A

r times
• (this is well defined because the Boolean product of matrices
is associative.)
• We also define A[0] to be In
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Zero-One Matrices
• Example 11: Let
0 0 1  .
A  1 0 0
1 1 0
Find A[n] for all positive integers n.
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