Matrices

1. Matrices

2. Matrices • A matrix is a rectangular array of objects (usually numbers) arranged in mhorizontal rows and n vertical columns. • A matrix with m rows and ncolumns is called an mx nmatrix. • The plural of matrix is matrices. • The ithrow of A is the 1× n matrix [ai1, ai2,…, ain], 1≤ i≤ m. The jthcolumn of A is the m × 1 matrix: , 1≤ j ≤ n. Example: The matrix is a 3 x 2 matrix.

3. Matrices • We refer to the element in the ithrow and jthcolumn of the matrix Aas aijor 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 orderis n. • Two matrices are equalif they have the same number of rows and the same number of columns and the corresponding entries in every position are equal. 22

4. Example 1 • ThenA 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

5. Diagonal Matrix • Asquare 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

6. 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

7. Cont’d • The following matrix gives the airline distance between the cities indicated

8. Matrix Equality • Two m x n matrices A = [aij] and B = [bij] are said to be equalif aij=bij, 1 ≤ i ≤ m, 1 ≤ j ≤ n; that is, if corresponding elements in every position are the same.

9. Cont’d Then A = B if and only if x=-3, y=0, and z=6 If

10. The sum of two matrices of the same size is obtained by adding elements in the corresponding positions. Matrices of different sizes cannotbe added. Matrix Arithmetic DEFINITION 3: Let A = [aij] and B = [bij] be m x nmatrices. The sumof A and B, denoted by: A + B, is the m x nmatrix that has aij + bijas its ( i, j )thelement. In other words, A + B = [aij + bij].

11. Example 1 Example 2

12. Zero Matrix • A matrix allof whose entries are zerois called: azero matrix and is denoted by 0 • Each of the following is Zero matrix:

13. Properties of Matrix Addition • A + B = B + A • (A + B) + C = A + (B + C) • A + 0 = 0 + A = A

14. The product of the two matrices is not defined when the number of columnsin the firstmatrix and the number of rows in the second matrixis not the same. Matrices Production DEFINITION 4: Let A be an m x k matrix and B be a k x nmatrix. The product of A and B, denoted by AB, is the m x n matrix with its (i, j )thentry equal to the sum of the productsof the corresponding elements from the I throw of A and the j th column of B. In other words, if AB = [cij], then cij = ai1 b1j+ ai2 b2j+ … + aikbkj. NOTE: Matrix multiplication is not commutative!

15. Cont’d

16. Cont’d • Example: Let A = and B = Find AB if it is defined. AB = 3×2 4×3 4×2

17. Cont’d • Example: 2×3 3×4 2×4

18. 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 =

19. 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: • BA may not be defined; we may have n ≠ m • 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 • AB and BA may both the same size, but not equal as matrices AB ≠ BA • AB = BA

20. Basic Properties of Multiplication • The basic properties of matrix multiplication are given by the following theorem: • A(BC) = (AB)C • A(B + C)= AB + AC • (A + B)C = AC + BC

21. 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 = ImA = A

22. p times Powers of Matrices • Powers of square matrices can be defined. • If A is an nn square matrix and p  0, we have Ap AAA ··· A • A0nxn= Insquare matrix to the zero power is identity matrix. • Example:

23. Powers of Matrices cont. • If p and q are nonnegative integers, we can prove the following laws of exponents for matrices: • ApAq = 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. 23

24. Example: The transpose of the matrix is the matrix Example 2: Let , Find At. Transpose Matrices DEFINITION 6: Let A = [aij] be an m x n matrix. The transpose of A, denoted by At, is the n x mmatrix 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.

25. Properties for Transpose • If A and B are matrices, then

26. Example: The matrix is symmetric. Example: Symmetric Matrices DEFINITION 7: A square matrix A is called symmetricif A = At. Thus A = [aij] is symmetric if aij = aji for all i and j with 1 <= i <= n and 1 <= j <= n.

27. Symmetric Matrices • Which is symmetric? ABC

28. 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 • The operations on zero-one matrices is based on the Boolean operations v and ^, which operate on pair of bits.

29. Boolean Matrix Operations- OR • Let A = [aij] and B = [bij] be m x n Boolean matrices. • We define A v B = C = [ Cij], the joinof A and B, by 1 if aij = 1 orbij = 1 Cij = 0 if aijandbij are both 0

30. Example • Find the join of A and B: A = 1 0 1 B = 0 1 0 0 1 0 1 1 0 The joinbetween A and B isA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

31. Boolean Matrix Operations- Meet • We define A ^ B = C = [ Cij], the meet of A and B, by 1 if aijandbij are both 1 Cij = 0 if aij = 0 orbij = 0 • Meet & Join are the same as the addition procedure • Each element with the correspondingelement in the other matrix • Matrices have the same size

32. 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

33. Boolean PRODUCT A⊙B • The Boolean product of A and B, denoted, is the m x n Boolean matrix defined by: 1 if aik = 1 andbkj = 1 for some k, 1 ≤ k ≤ p Cij 0 otherwise • The Boolean product of Aand 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

34. Example Find the Boolean product of A and B: A 3x2= 1 0 B 2x3 = 110 0 1011 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 A⊙B

35. Boolean Operations Properties • If A, B, and C are Boolean Matrices with the same sizes, then • A v B = B v A • A ^ B = B ^ A • (A v B) v C = A (B v C) • (A ^ B) ^ C = A ^ (B ^ C)

36. Boolean Powers • For a square zero-one matrix A, and any k  0,the kthBoolean power of A is simply the Boolean product of k copies of A. A[k]  A⊙A⊙…⊙A k times

37. 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 37

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