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## Lecture 2.6: Matrices*

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**Lecture 2.6: Matrices***CS 250, Discrete Structures, Fall 2011 Nitesh Saxena Lecture 2.6 -- Matrices**Course Admin**• Mid-Term 1 on Thursday, Sep 22 • 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 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 Lecture 2.6 -- Matrices**Outline**• Matrix • Types of Matrices • Matrix Operations Lecture 2.6 -- Matrices**Matrix – what it is?**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 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 • … Lecture 2.6 -- Matrices**Types of Matrices**Square Matrix Number of rows = number of columns Which one(s)of the following is(are) squarematrix(ces)? Where is the main diagonal? 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” 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 Lecture 2.6 -- Matrices**Equality of Matrices**Find a, b, c, and d Find a, b, c, k, m, x, y, and z Equal Matrices - Work this out If 2. If 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} 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. Lecture 2.6 -- Matrices**Multiplying two Matrices**• 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 Matrices Products Lecture 2.6 -- Matrices**Multiplying two Matrices**Matrices Products – Possible outcomes Lecture 2.6 -- Matrices**Multiplying two Matrices**Show that AB is NOT BA (this means that matrix multiplication is not commutative) Matrices Products – Work this out Let 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} Lecture 2.6 -- Matrices**Matric Transpose**Compute (BA)T : Compute AT(D + F) Transposition Matrix – Work this out 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 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 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 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 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: Lecture 2.6 -- Matrices**Boolean Matrices**Lecture 2.6 -- Matrices**Boolean Matrices**Work this out Lecture 2.6 -- Matrices**Today’s Reading**• Rosen 2.6 Lecture 2.6 -- Matrices