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.

312011

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.

1312

22

061

0236123

3

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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211031143

043322101

Example 2

Example 1

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252313244

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 =

220013112401

031142

2813798414

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4×3 3×2

4×2

Cont’d• Example:

130102020110

302110

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2×3 3×4 2×4

311231501

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 =

1211

1112

3523

2334

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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

010001

if 0 if 1

jiji

In

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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

2334

1223

0112

0112

0112

0112

0112 3

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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.

654321

635241

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210

312A

231102

TA

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.

010101011

Symmetric Matrices

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Symmetric Matrices• Which is symmetric?

A B C

111111

213101312

211120103

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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 10 0 11 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 = 1Cij = 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 1Cij = 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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ExampleFind 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 Cij = 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,

thena. A v B = B v Ab. A ^ B = B ^ Ac. (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