Lecture 5: Matrices. Dheeraj Kumar Singh 07CS1004 Teacher: Prof. Niloy Ganguly Department of Computer Science and Engineering IIT Kharagpur

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1 Lecture 5: Matrices Dheeraj Kumar Singh 07CS1004 Teacher: Prof. Niloy Ganguly Department of Computer Science and Engineering IIT Kharagpur 29 th July, 2008 Types of Matrices Matrix Addition and Multiplication Boolean Matrix Operations 1

2 1 MATRICES 1 Matrices A Matrix is a rectangular array of numbers arranged in m horizontal rows and n vertical columns. a 11 a a 1n a 21 a a 2n a n1 a n2... a nn Here we say that A is of order m n A can also be written as The different types of Matrices are [a ij ] 1. Square Matrix: The number of rows and number of columns in a Square Matrix are equal. If a matrix A is n n it is said to be a square Matrix of order n. The entries a 11, a 22, a 33,..., a nn form the Main Diagonal of A. For example if Then A is a square Matrix of order Diagonal Matrix: A Square Matrix [a ij ]is said to be a Diagonal Matrix if every element off its Main Diagonal is 0. Example : A 1 = A 2 =

3 1 MATRICES 3. Zero Matrix: A Matrix is said to be a Zero Matrix if all its entries are 0. A Zero Matrix is denoted by 0 Example: A 1 = A 2 = [ 4. Identity Matrix: An n n diagonal matrix whose all the diagonal elements are 1 is called an Identity Matrix of order n. I n = ] Symmetric Matrix: A square matrix [a ij ] of order n is said to be a symmetric matrix if a ij = a ji for all i, j such that 1 i, j n. 3

4 2 ADDITION OF MATRICES: 2 Addition of Matrices: Two matrices [a ij ] and B = [b ij ] can be added only if they are of the same order. If A and B are m n matrices then the matrix C = A + B is defined as c ij = a ij + b ij for 1 <= i <= m, 1 <= j <= n Example : B = A + B = Some of the properties about Matrix Addition are A + B = B + A (A + B) + C = A + (B + C) A + 0 = 0 + A 4

5 3 PRODUCT OF MATRICES 3 Product of Matrices The product of two Matrices [a ij ] of order m 1 n 1 and B = [b ij ] of order m 2 n 2 is defined only if n 1 = m 2. Let A be a m p matrix and B be a p n matrix. Then let c ij = a i1 b ij + a i2 b 2j a ip b pj C = AB Then the matrix C = [c ij ] is defined by ie. c ij = p a ik b kj The order of C is m n. To get c ij we select the i th row of A and j th column of B, then multiply their corresponding elements and then take their sum as illustrated in the figure below: k=1 Multiplication of Matrices 5

6 3 PRODUCT OF MATRICES If AB is defined BA may not necessarily be defined. For BA to be defined m must be equal to n. The order of BA then is p p. Even if BA is defined they may not be of the same order if p is not equal to m. They are of the same order only if p = m. In this case also BA is not necessarily equal to AB. AB MAY be equal to BA. The properties associated with Matrix Multiplication are: A(BC) = (AB)C A(B + C) = AB + AC (A + B)C = AC + BC If A is a square matrix of order n If p > 0, A p = A.A.A...A p times and A 0 = I n Some observations about multiplication of Matrices are : A is considered a matrix of order m n I m AI n = A A p A q = A p+q (A p ) q = A pq (AB) p may be equal to A p B p If AB = BA, then (AB) p = A p B p Multiplicative Inverse A is a square matrix of order n. Let B be a square matrix also of order n such that AB = B I n. Then B is called the multiplicative inverse of A. Inverse of a matrix A exists if Det(A) 0. 6

7 4 TRANSPOSE OF A MATRIX 4 Transpose of a Matrix If [a ij ] is a m n matrix, then the matrix A T = [a T ij] is a n m matrix where a T ij = a ji, for 1 i m, 1 j n. A T is called the Transpose of A. In short transpose of A can be obtained by interchanging the rows and columns. Example: Then A T = Some of the basic properties related with transpose of a Matrix are: (A T ) T = A (A + B) T = A T + B T (AB) T = B T A T For a symmetric matrix A, A T 7

8 5 BOOLEAN MATRIX OPERATIONS 5 Boolean Matrix Operations A Boolean Matrix(also called a bit matrix)has its entries either 0 or 1. Three operators defined on them are, and. Let [a ij ] and B = [b ij ] be m n Boolean Matrices. Let C = A B = [c ij ]. C is then said to be the join of A and B. C is also a m n Matrix. { 1 if aij = 1 or b c ij = ij = 1 0 if a ij = 0 and b ij = 0 Let D = A B = [d ij ]. D is then said to be the meet of A and B. D is also a m n Matrix. { 1 if aij = 1 and b d ij = ij = 1 0 if a ij = 0 or b ij = 0 The above operations are ONLY POSSIBLE if the two matrices have the same size. The Boolean Product represented by A B is quite different from the above two operations. Let A be a boolean matrix of order a 1 a 2 and B a boolean matrix of order b 1 b 2. A B is defined only if a 2 = b 1. So let A have an order of m p and B have an order of p n Let E = A B = [e ij ]. E is then said to be the Boolean Product of A and B. E is a m n Matrix. { 1 if aik = 1 and b e ij = kj = 1 for some k, 1 k p 0 otherwise For e ij we select the i th row of A and j th column of B. If even one of the corresponding entries are both 1, the result is 1, else the result is 0(as illustrated in the figure below). 8

9 5 BOOLEAN MATRIX OPERATIONS Boolean Product of Matrices Some properties of Boolean Operations are: Let A, B, C be boolean matrices of compatible size. Then : A B = B A A B = B A (A B) C = A (B C) (A B) C = A (B C) A (B C) = (A B) (A C) A (B C) = (A B) (A C) (A B) C = A (B C) 9

10 5 BOOLEAN MATRIX OPERATIONS Multiplication of Boolean Matrices A, B are boolean matrices. The compatibility rules for AB to be defined remain the same as that for Boolean Product. So let A have an order of m p and B have an order of p n Let F = AB = [f ij ]. F is a m n Matrix. f ij = ( p a ik b kj ) mod 2 k=1 If we go on multiplying a bit matrix by itself, we sometimes get back the original matrix. Example: [ ] = A 1 [ ] = A 2 = = A 2 = A 3 = = A 3 = A 1 = A 2 = A 4 = A 1 = A 5 = A 2 = A 2...And so on So these three matrices form a cycle. This is a cycle of size 3. The largest cycle that n n boolean matrices can form is of size 2 n 1. Let us consider the other 2 2 Boolean Matrices. [ ] A 4 = A 2 4 = [ ] [ ] A 5 = [ ] A 2 5 = 10

11 5 BOOLEAN MATRIX OPERATIONS A 6 = A 2 6 = [ ] A 7 = [ ] A 2 7 = no cycle is formed [ ] A 8 = [ ] A 2 8 = no cycle is formed [ ] A 9 = [ ] A 2 9 = A 10 = A 2 10 = [ ] A 11 = [ ] A 2 11 = 11

12 5 BOOLEAN MATRIX OPERATIONS [ ] A 12 = [ ] A 2 12 = [ ] A 13 = A 2 13 = a cycle of 2 A 14 = A 2 14 = a cycle of 2 [ ] A 15 = A 2 15 = a cycle of 2 [ ] A 16 = [ ] A 2 16 = no cycle formed The largest cycle has a size of 3, which is consistent with the rule(2 2 1 = 3) If we take a 2 2 matrix and go on multiplying it with 2 1 matrices, we get cycles here also. Example: [ ] 12

13 [ 1 A 1 [ 1 A [ 0 A 1 5 BOOLEAN MATRIX OPERATIONS ] [ ] 1 = 0 ] [ ] 0 = ] [ ] 1 = 1 a cycle of 3 [ ] [ The cycles are represented pictorially as shown: ] So these form a cycle of 3 and written as [1(1), 1(3)] Another example [ ] [ ] [ ] [ ] a cycle of 2 13

14 [ 0 A 1 5 BOOLEAN MATRIX OPERATIONS ] = [ 0 1 [ ] [ The cycles are represented pictorially as shown: ] ] So these form a cycle of 2 and 2 cycles of 1 written as [1(2), 2(1)] Another Example: [ ] [ ] [ ] [ ] [ ] [ ] The cycles are represented pictorially as shown: 14

15 5 BOOLEAN MATRIX OPERATIONS Here we find unreachable states. In the loop we never get back to (1, 0)and(0, 1). 15