MATRICES

Matrices• A matrix is a rectangular array of objects (usually numbers)

arranged in m horizontal rows and n vertical columns.• A matrix with m rows and n columns is called an

m x n matrix.

• The plural of matrix is matrices.

• The ith row of A is the 1× n matrix [ai1, ai2,…, ain], 1≤ i ≤ m. The jth column of A is the m × 1 matrix: , 1≤ j ≤ n.

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Example: The matrix is a 3 x 2 matrix.

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20

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Matrices

• We refer to the element in the ith row and jth column of the matrix A as aij or as the (i, j) entry of A, and we often write it as A= [aij ].

• A matrix with the same number of rows as columns is called square matrix, whose order is n.

• 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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1222

061

023

61

23

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Then A is 2 x 3 with a12 = 3 and a23 = 2,B is 2 x 2 with b21 = 4, C is 1 x 4, D is 3 x 1, and E is 3 x 3

Example 1

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A square matrix A = [aij ] for which every entry off the main diagonal is zero, that is, aij = 0 for i ≠ j, is called a diagonal matrix

Example

Diagonal Matrix

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Example of Matrix applications

• Matrices are used in many applications in computer

science, and we shall see them in our study of relations

and graphs.

• At this point, we present the following simple application

showing how matrices can be used to display data in a

tabular form

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Cont’d• The following matrix gives the airline distance between

the cities indicated

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Matrix Equality

• Two m x n matrices A = [aij ] and B = [bij] are said to be

equal if aij = bij , 1 ≤ i ≤ m, 1 ≤ j ≤ n;

that is, if corresponding elements in every position are the

same.

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Then A = B if and only if x=-3, y=0, and z=6

Cont’d

If

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• The sum of two matrices of the same size is obtained by

adding elements in the corresponding positions.

• Matrices of different sizes cannot be added.

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].

Matrix Arithmetic

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211

031

143

043

322

101

Example 2

Example 1

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252

313

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Zero Matrix• A matrix all of whose entries are zero is called:

a zero matrix and is denoted by 0

• Each of the following is Zero matrix:

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Properties of Matrix Addition

• A + B = B + A

• (A + B) + C = A + (B + C)

• A + 0 = 0 + A = A

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• The product of the two matrices is not defined when the

number of columns in the first matrix and the number of

rows in the second matrix is not the same.

DEFINITION 4:

Let A be an m x k matrix and B be a 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 I th row of A and the j th column of B.

In other words, if AB = [cij], then

cij = ai1 b1j + ai2 b2j + … + aik bkj.

Matrices Production

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NOTE: Matrix multiplication is not commutative!

Cont’d

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Cont’d• Example:

Let A = and B =

Find AB if it is defined.

AB =

220

013

112

401

03

11

42

28

137

98

414

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4×3 3×2

4×2

Cont’d• Example:

1301

0202

0110

302

110

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2×3 3×4 2×4

31123

1501

Matrices• If A and B are two matrices, it is not necessarily true that AB and BA

are the same.

• E.g. if A is 2 x 3 and B is 3 x 4, then AB is defined and is 2 x 4, but BA is not defined.

• Even when A and B are both n x n matrices, AB and BA are not necessarily equal.

• Example:

Let A 2x2 = and B 2x2=

Does AB = BA?

Solution:

AB = and BA =

12

11

11

12

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23

23

34

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Properties of Multiplication

• If A = m x p matrix, and B is a p x n matrix, then AB can be

computed and is an m x n matrix.

• As for BA, we have four different possibilities:

1. BA may not be defined; we may have n ≠ m

2. BA may be defined if n = m, and then BA is p x p, while AB

is m x m and p ≠ m. Thus AB and BA are not equal

3. AB and BA may both the same size, but not equal as

matrices AB ≠ BA

4. AB = BA19

Basic Properties of Multiplication• The basic properties of matrix multiplication are given by

the following theorem:

1. A(BC) = (AB)C

2. A(B + C)= AB + AC

3. (A + B)C = AC + BC

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Identity Matrix

• The n×n diagonal matrix all of whose diagonal elements are 1

and 0’s everywhere else, is called the identity matrix of order

n, denoted by In.

• A In = A

• Multiplying a matrix by an appropriately sized identity matrix

does not change this matrix.

• In other words, when A is an m x n matrix, we have A In = Im A = A

100

010

001

if 0

if 1

ji

jiIn

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Powers of Matrices

• Powers of square matrices can be defined.

• If A is an nn square matrix and p 0, we have

Ap AAA ··· A

• A0nxn = In square matrix to the zero power is identity matrix.

• Example:

p times

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12

23

01

12

01

12

01

12

01

12

01

123

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Powers of Matrices cont.

• If p and q are nonnegative integers, we can prove the following laws of exponents for matrices:

• Ap Aq = Ap+q

• (Ap)q =Apq

• Observe that the rule (AB)p =ApBp does not hold for square matrices unless AB = BA.

• If AB = BA, then (AB)p =ApBp.

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• Example:

The transpose of the matrix is the matrix

• Example 2: • Let , Find At.

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.

654

321

63

52

41

24

210

312A

23

11

02TA

Transpose Matrices

Properties for Transpose

• If A and B are matrices, then

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• Example:

The matrix is symmetric.

• Example:

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.

010

101

011

Symmetric Matrices

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Symmetric Matrices• Which is symmetric?

A B C

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11

11

213

101

312

211

120

103

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Boolean Matrix Operation

A matrix with entries that are either 0 or 1 is called a

Boolean matrix or zero-one matrix.

• 0 and1 representing False & True respectively.

Example:

1 0 1

0 0 1

1 1 0

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• The operations on zero-one matrices is based on the

Boolean operations v and ^, which operate on pair of bits.

Boolean Matrix Operations- OR

• Let A = [aij] and B = [bij] be m x n Boolean matrices.

1. We define A v B = C = [ Cij], the join of A and B, by

1 if aij = 1 or bij = 1

Cij = 0 if aij and bij are both 0

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Example• Find the join of A and B:

A = 1 0 1 B = 0 1 0

0 1 0 1 1 0

The join between A and B is A B =

= 1 v 0 0 v 1 1 v 0 = 1 1 1

0 v 1 1 v 1 0 v 0 1 1 0

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Boolean Matrix Operations- Meet • We define A ^ B = C = [ Cij], the meet of A and B, by

1 if aij and bij are both 1

Cij = 0 if aij = 0 or bij = 0

• Meet & Join are the same as the addition procedure Each element with the corresponding element in the other

matrixMatrices have the same size

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Example

Find the meet of A and B:

A = 1 0 1 B = 0 1 0

0 1 0 1 1 0

A v B = 1 ^ 0 0 ^ 1 1 ^ 0 = 0 0 0

0 ^ 1 1 ^ 1 0 ^ 0 0 1 0

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Boolean PRODUCT• The Boolean product of A and B, denoted,

is the m x n Boolean matrix defined by: 1 if aik = 1 and bkj = 1 for some k, 1 ≤ k ≤ p

Cij 0 otherwise

• The Boolean product of A and B is like normal matrix product, but using instead + and using instead .

Procedure:• Select row i of A and column j of B, and arrange them side by side• Compare corresponding entries. If even a single pair of

corresponding entries consists of two 1’s, then C ij = 1, otherwise Cij = 0

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A⊙B

ExampleFind the Boolean product of A and B:

A 3x2= 1 0 B 2x3 = 1 1 0

0 1 0 1 1

1 0

3x3 = (1 ^ 1) v (0 ^ 0) (1 ^ 1) v (0 ^ 1) (1 ^ 0) v (0 ^ 1)

(0 ^ 1) v (1 ^ 0) (0 ^ 1) v (1 ^ 1) (0 ^ 0) v (1 ^ 1)

(1 ^ 1) v (0 ^ 0) (1 ^ 1) v (0 ^ 1) (1 ^ 0) v (0 ^ 1)

1 1 0

= 0 1 1

1 1 0

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A⊙B

Boolean Operations Properties• If A, B, and C are Boolean Matrices with the same sizes,

then

a. A v B = B v A

b. A ^ B = B ^ A

c. (A v B) v C = A (B v C)

d. (A ^ B) ^ C = A ^ (B ^ C)

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Boolean Powers

• For a square zero-one matrix A, and any k 0, the kth

Boolean power of A is simply the Boolean product of k

copies of A.

A[k] A⊙A⊙…⊙A

k times

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Example

• Find A [n]for all positive integers n .

• Solution: We find that

We also find that :

Additional computation shows that

We can notice that A [n] = A [5] for all positive integers n with n ≥ 5

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Any Question• Refer to chapter 3 of the book for further reading