CS 250, Discrete Structures, Fall 2011 Nitesh Saxena Lecture 2.6: Matrices* CS 250, Discrete Structures, Fall 2011 Nitesh Saxena 9/15/2011 Lecture 2.6 -- Matrices

Course Admin Mid-Term 1 on Thursday, Sep 22 No lecture on Sep 20 In-class (from 11am-12:15pm) Will cover everything until the lecture on Sep 15 No lecture on Sep 20 As announced previously, I will be traveling to Beijing to attend and present a paper at the Ubicomp 2012 conference This will not affect our overall topic coverage This will also give you more time to prepare for the exam 9/15/2011 Lecture 2.6 -- Matrices

Course Admin HW2 has been posted – due Sep 30 Covers chapter 2 (lectures 2.*) Start working on it, please. Will be helpful in preparation of the mid-term HW1 grading delayed a bit TA/grader was sick with chicken pox Trying to finish as soon as possible HW1 solution has been released 9/15/2011 Lecture 2.6 -- Matrices

Outline Matrix Types of Matrices Matrix Operations 9/15/2011 Lecture 2.6 -- Matrices

Matrix – what it is? A = {aij}, where i = 1, 2, …, m and j = 1, 2,…, n An array of numbers arranged in m horizontal rows and n vertical columns. We say that A is a matrix m x n. (Dimension of matrix) . A = {aij}, where i = 1, 2, …, m and j = 1, 2,…, n 9/15/2011 Lecture 2.6 -- Matrices

Examples Grades obtained by a set of students in different courses can be represented a matrix Average monthly temperature at a set of cities can be represented as a matrix Facebook friend connections for a given set of users can be represented as a matrix … 9/15/2011 Lecture 2.6 -- Matrices

Types of Matrices Square Matrix Number of rows = number of columns Which one(s)of the following is(are) square matrix(ces)? Where is the main diagonal? 9/15/2011 Lecture 2.6 -- Matrices

Types of Matrices Diagonal Matrix “a square matrix in which entries outside the main diagonal area are all zero, the diagonal entries may or may not be zero” 9/15/2011 Lecture 2.6 -- Matrices

Equality of Matrices Two matrices are said to be equal if the corresponding elements are equal. Matrix A = B iff aij = bij Example: If A and B are equal matrices, find the values of a, b, x and y 9/15/2011 Lecture 2.6 -- Matrices

Equality of Matrices Equal Matrices - Work this out If 2. If Find a, b, c, and d Find a, b, c, k, m, x, y, and z 9/15/2011 Lecture 2.6 -- Matrices

Adding two Matrices Matrices Summation The sum of the matrices A and B is defined only when A and B have the same number of rows and the same number of columns (same dimension). C = A + B is defined as {aij + bij} 9/15/2011 Lecture 2.6 -- Matrices

Adding Two Matrices Matrices Summation – work this out a) Identify the pair of which matrices between which the summation process can be executed b) Compute C + G, A + D, E + H, A + F. 9/15/2011 Lecture 2.6 -- Matrices

Multiplying two Matrices Matrices Products Steps before Find out if it is possible to get the products? Find out the result’s dimension Arrange the numbers in an easy way to compute – avoid confusion 9/15/2011 Lecture 2.6 -- Matrices

Multiplying two Matrices Matrices Products – Possible outcomes 9/15/2011 Lecture 2.6 -- Matrices

Multiplying two Matrices Matrices Products – Work this out Let Show that AB is NOT BA (this means that matrix multiplication is not commutative) 9/15/2011 Lecture 2.6 -- Matrices

Matrix Transpose Transposition Matrix A matrix which is formed by turning all the rows of a given matrix into columns and vice-versa. The transpose of matrix A is written AT, and AT = {aji} 9/15/2011 Lecture 2.6 -- Matrices

Matric Transpose Compute (BA)T : Compute AT(D + F) Transposition Matrix – Work this out Compute (BA)T : Compute AT(D + F) 9/15/2011 Lecture 2.6 -- Matrices

Symmetric Matrix Symmetrical Matrix A is said to be symmetric if all entries are symmetrical to its main diagonal. That is, if aij = aji 9/15/2011 Lecture 2.6 -- Matrices

Boolean Matrices Boolean Matrix and Its Operations Boolean matrix is an m x n matrix where all of its entries are either 1 or 0 only. There are three operations on Boolean: Join by Meet Boolean Product 9/15/2011 Lecture 2.6 -- Matrices

Boolean Matrices Boolean Matrix and Its Operations – Join By Given A = [aij] and B = [bij] are Boolean matrices with the same dimension, join by A and B, written as A  B, will produce a matrix C = [cij], where cij = aij  bij 9/15/2011 Lecture 2.6 -- Matrices

Boolean Matrices Boolean Matrix and Its Operations – Meet Meet for A and B, both with the same dimension, written as A  B, will produce matrix D = [dij] where dij = aij  bij 9/15/2011 Lecture 2.6 -- Matrices

Boolean Matrices Boolean Matrix and Its Operations – Boolean Products If A = [aij] is an m x p Boolean matrix, and B = [bij] is a p x n Boolean matrix, we can get a Boolean product for A and B written as A ⊙ B, producing C, where: 9/15/2011 Lecture 2.6 -- Matrices

Boolean Matrices 9/15/2011 Lecture 2.6 -- Matrices

Boolean Matrices Work this out 9/15/2011 Lecture 2.6 -- Matrices

Today’s Reading Rosen 2.6 9/15/2011 Lecture 2.6 -- Matrices