Homework 1, Math 307-101, Fall 2013 (Final Version)

Due Monday, September 16, 2013

Review: know how to diagonalize matrices and review powers of matrices.

In this homework we work with some 2 x 2 examples. Homework 2 will do this with larger matrices obtained by card shuffling schemes.

New: Use Matlab to do the work for you.

Note: Problems 2-6 are simple and short Matlab computations; collectively will probably take you less time than Problem 1.

Exam Note: On exams, I will never ask you to compute eigenvalues via characteristic polynomials (I will supply the relevant eigenvalues or rely on theorems we cover in this course, e.g., an irreducible Markov matrix has 1 as its dominant eigenvalue). You will need to understand how to diagonalize a matrix (when possible) given an eigenbasis.

Three problems indicated "[A]" will be the focus of the in-depth assessment. Feedback on the other problems will be given as time permits, but do not count for your homework grade.

  1. [A] REVIEW: A=[1/2,1/3;1/2,2/3]. Compute Delta(t), the characteristic polynomial of A. Is A diagonalizable? If yes, find P such P^(-1) A P is diagonal. Hint: A has eigenvalues 1,1/6. This is similar to 16.69 of 3,000 Solved Problems in Linear Algebra (with different numbers); note that 3,000 Solved ... writes the column vector v1 as (2,-1); in class we may write (2,-1)^T to emphasize that this is a column vector.

  2. Let A be as above. Use Matlab (or the software of your choice) to compute the value of A, A^2, A^3, A^5, A^10, A^100. What do you think is the limit of A^n as n tends to infinity?

  3. Issue the Matlab command "format long" and recompute A^5, A^10, A^100 with A as above. Note that in Matlab, you are doing the exact same computation; in this exercise you are merely printing things out in double precision.

  4. [A] Let A be as above.
    1. Issue the Matlab command: eigs(A) to see the eigenvalues.
    2. Issue the Matlab command: [P,D] = eig(A).
    3. Does this last calculation solve the first homework problem? Issue the Matlab command: inv(P) * A * P to answer this question, comparing and constrasting with your solution to Problem 1 of this homework set.

  5. Same Matlab question as previously, with A = [2,2;1,3], still in "format long". This is similar to 16.69 in 3,000 Solved ... Do you notice a round-off error in double precision?

  6. [A] Let A=[1,1;1,0]. Privately compute A,A^2,...,A^15; by "privately" I mean DO NOT HAND IN THIS COMPUTATION; submit only A^2,A^3,A^4,A^5,A^7,A^9. Look at this introduction to Fibonacci numbers, or look up the Fibonacci entry of Wikipedia, and generate the first 10 Fibonacci numbers; you may do this by hand or computer; you can just download fibonacci.m from the exm toolkit, place this file in any directory, and select "set path" from Matlab to include this directory, and type "fibonacci(10)". Based on your experiments, describe the 2 x 2 matrix A^n, for any positive integer n, as a formula in terms of Fibonacci numbers (make an educated guess based on your numerical experiments).