Computer Project #2 (Matrix Operations)

Transcription

1 Math 0280 Introduction to Matrices and Linear Algebra Fall 2006 Computer Project #2 (Matrix Operations) SCHEDULE: This assignment is due in class on Monday, October 23, One submission per group is sufficient. A printout of the completed m-file must be submitted. OVERVIEW: In the second exploration, we consider two applications of matrix multiplication. After reviewing how to carry out matrix multiplication and finding inverses in MAT- LAB, we discuss 1) Markov chains and 2) graphs. MATLAB CONCEPTS: Suppose that we are given two matrices A and B such that A = and B = To multiply A and B in MATLAB, type PROMPT>> A=[1,2,-3;4,5,6;,7,8,9]; <enter> PROMPT>> B=[1,2;3,4;5,6]; <enter> PROMPT>> A*B <enter> to output the product AB To find the power A k of a square matrix A, type either A*A* }{{ *A} or A^k. For example, k times PROMPT>> A^3 <enter> gives the value of A To find the inverse A 1 of a square matrix A, type either inv(a) or A^-1. For example, PROMPT>> inv(a) <enter> displays the inverse A

2 2 MARKOV CHAINS: (See pp in the text) Markov chains model the process of making decisions according to a finite set of probabilities. Let s illustrate with an example. Suppose that the same 600 people eat out every Friday at either Restaurant 1 (Mexican), Restaurant 2 (Chinese), or Restaurant 3 (Italian). The diagram represents the probability that a person will eat at Restaurant x, if he ate at Restaurant y the week before. For example, if a person ate Italian the previous week, then the next Friday he is 10% likely to eat Italian, 40% to eat Mexican, and 50% to eat Chinese. As a matrix T, called the transition matrix of the Markov chain, the diagram becomes T = to 1 to to 3 from where the (i, j) entry of T represents the probability that eating at restaurant j last week, a person will eat at restaurant i next Friday (i.e., the probability of going from j to i). Observe that the each column of T sums to 1; by definition, this occurs in any Markov chain. Suppose that one week 100 people eat Mexican, 200 people eat Chinese, and 300 people eat Italian, say x 1 = [100, 200, 300] T. What happens the following week? Using T, we calculate x 2 = T x 1 = = that 210 people eat Mexican, 220 eat Chinese, and 170 eat Italian. How many people eat at each weekend the following week? Again, we use T and calculate 2 x 3 = T x 2 = T (T x 1 ) = T 2 x 1 = = that 197 people eat Mexican, 192 people eat Chinese, and 211 people eat Italian. Inductively, if we repeat this process, the sequence x 1, T x 1, T 2 x 1, T 3 x 1,... forms a Markov chain.

3 GRAPHS: (See pp in the text) By definition, a graph G = (V, E) is an object with a set of vertices V (points) connected by a set of edges E (lines). Let s look at an example. 3 The diagram above represents a graph with 4 vertices {1, 2, 3, 4} and 4 edges {13, 23, 24, 34}. Given a graph G with n vertices, we define its adjacency matrix to be the n n matrix whose (i, j) entry equals 1, if ij is an edge in the graph, equals 0, otherwise. For our example above, A = is the corresponding adjacency matrix. Observe that A is a symmetric matrix (i.e., A = A T ). In fact, the adjacency matrix of any graph is symmetric, because if there is an edge between vertices i and j, then there is also (in fact, the same) edge between j and i. Let s consider motion in the graph. Informally, we think about walking along edges of a graph to form a path between two points. More exactly, we define a path between two vertices i and j to be a sequence of edges iv 2, v 2 v 3, v 3 v 4,, v k 1 v k, v k j in the graph, where v 2,, v k are arbitrary vertices; the length of the path is the number of terms in the sequence. In our running example, there are two paths between 1 and 4. First, 13, 34 is a path of length 2. Second, 13, 32, 24 is a path of length 3. To find all the paths of length 2 in a graph, we consider the square of its adjacency matrix. In the running example, for instance, A 2 = Now, the (i, j) entry of A 2 is exactly the number of length 2 paths between i and j. (STOP! Make sure that you check this yourself!) To find the number of length 2 paths in our graph, then, it suffices to add the entries on and above the main diagonal of A 2. In this case here, there are exactly 13 paths of length 2. In general, the (i, j) entry of A k is exactly the number of length k paths between i and j.

4 4 ASSIGNMENT: (OUT OF 40 POINTS) Complete all three parts and submit project2.m. For each question, please provide any MATLAB commands that you use for computation. I. Exercise (8 Points) Let A, B, C and D be the following matrices: [ ] 1 7 [ ] A = , B =, C = , D = (a) Which of the following sixteen products exist?: A A, A B, A C, A D, B A, B B, B C, B D, C A, C B, C C, C D, D A, D B, D C, and D D. (b) Pick a product from part (a) that exists. Find its value. (c) Pick a product from part (a) that does not exist. Why does it not exist? (d) Find the inverse for each of A, B, C and D, or explain why it does not exist. II. Markov Chains United States Census (16 POINTS) The U.S. Census Bureau divides the United States into four geographical distinct regions: the Northeast, the Midwest, the South, and the West. According to the 2000 census, in the year between 1999 and 2000, people moved between the regions as follows: People Moving From Northeast Midwest South West People Moving To Northeast 98.85% 0.11% 0.18% 0.17% Midwest 0.15% 99.01% 0.42% 0.35% South 0.76% 0.57% 98.97% 0.78% West 0.24% 0.32% 0.43% 98.70% 2000 Population (in thousands) (a) Find the transition matrix for the population movement between regions above. (b) Which region had the greatest retention? Which had the greatest exodus? Explain. (c) Which region was the most popular destination of people changing regions? Explain. (d) Assuming zero net population change nationwide, form a Markov chain to predict the population of each region in Which regions lost population? Which gained? (e) Assuming zero net population change nationwide, predict the first year that the population in the Northeast will fall below 50 million people. (f) Are the predictions in part (d) and (e) accurate? Explain. III. Graphs Counting Paths (16 POINTS) For this part, refer to Graph A and Graph B. (a) How many vertices does Graph A have? How many edges? (b) Write down the adjacency matrix for Graph A. (c) How many paths of length 2 appear in Graph A?

5 5 (d) List all paths of length 3 that appear in Graph A. (e) A path between two vertices i and j is said to be minimal, if there is no other path between i and j with a smaller length. For example, in Graph B, 26, 63 is a minimal path between 2 and 3, yet 21, 15, 53 is not a minimal path. Identify the two minimal paths of length 4 in Graph A. Show that the matrix A + A 2 + A 3 + A 4 has no zero entries and conclude that every pair of vertices (i, j) in Graph A is connected by some path of length 4 or smaller. (f) Graph B is obtained by adding the edge 36 to Graph A. The result is that every pair of vertices (i, j) in Graph B is now connected by some path of length 3 or smaller. Similarly, we can form Graph C by adding an edge IJ to Graph B, so that every pair of vertices (i, j) in Graph C is connected by a path of length 2 or smaller. Write down the values for I and J.