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

2. 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
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Lecture Matrices

3. 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
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Lecture Matrices

4. Outline Matrix Types of Matrices Matrix Operations 9/15/2011
Lecture Matrices

5. 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
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Lecture Matrices

6. 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 …
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Lecture Matrices

7. 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?
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8. 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”
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9. 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
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10. 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
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Lecture Matrices

11. 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}
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Lecture Matrices

12. 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.
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Lecture Matrices

13. 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
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Lecture Matrices

14. Multiplying two Matrices
Matrices Products – Possible outcomes
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15. Multiplying two Matrices
Matrices Products – Work this out Let
Show that AB is NOT BA (this means that matrix multiplication is not commutative)
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16. 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}
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17. Matric Transpose Compute (BA)T : Compute AT(D + F)
Transposition Matrix – Work this out
Compute (BA)T :
Compute AT(D + F)
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18. Symmetric Matrix
Symmetrical Matrix A is said to be symmetric if all entries are symmetrical to its main diagonal. That is, if aij = aji
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19. 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
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Lecture Matrices

20. 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
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Lecture Matrices

21. 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
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Lecture Matrices

22. 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:
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23. Boolean Matrices
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24. Boolean Matrices
Work this out
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25. Today’s Reading
Rosen 2.6
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Lecture Matrices