CS 285- Discrete Mathematics

1. CS 285- Discrete Mathematics Lecture 11

2. Section 3.8 Matrices • Introduction • Matrix Arithmetic • Transposes and Power of Matrices • Zero – One Matrices • Boolean Product of Matrices Matrices

3. Introduction • Definition: a matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an mx n matrix. • A matrix with the same number of rows and columns is a square matrix • Two matrices are equal if they have the same number of rows and columns and the corresponding entries in every position are equal. Matrices

4. Matrix Arithmetic – Addition • Let A = [ aij] and B = [bij] be mxn matrices. The sum of A and B , denoted by A + B, is the m x n matrix that has aij +bijas its (i,j)thelement. In other words, A + B = [ aij + bij ] • Ex. Matrices

5. Matrix Arithmetic - Product • 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 products of the corresponding elements from the ith row of A and the jth column of B. • AB = [cij], cij = ai1 b1j + ai2 b2j + ….. + aikbkj • Ex. Matrices

6. Product –Example II Matrices

7. Identity Matrix • Definition: The identity matrix of order n is the n x n matrix In = [ δij], where δij= 1 if i = j and δij= o if i≠ j. Hence: In= • A In = Im A = A, where A is an m x n matrix Matrices

8. Powers of Matrices r times • The powers of matrices is the product of this matrix by it self for r times. • Let A be an m x n matrix • A0 = In, Ar = AAA … A Matrices

9. Transposes t • Definition: The transpose of the mxn matrix A, denoted by At, is the n x m matrix obtained by interchanging the rows and the columns of A. • If At = [ bij ], then bij = aji for i = 1,2, …,n and j = 1,2, … , m • Ex. The transpose of matrix A is Matrices

10. Symmetric Matrices • Definition: A square matrix A is called symmetric if A = At. Thus A = [ aij] is symmetric of aij= ajifor all iand j with 1 ≤ i≤ n and 1 ≤ j ≤ m • Ex. Matrices

11. Zero-One Matrices • Definition: A matrix with entries that are either 0 or 1. • Let A = [ aij ] and B = [ bij ] be m x n zero-one matrices. • Join of A and B is a zer0-one matrix with the (i,j)th entry ajibij. A B • Meet of A and B is a zer0-one matrix with the (i,j)th entry ajibij. A B Matrices

12. Examples Matrices

13. Matrices