### Question Description

This is an project due by Matlab for linear Algebra class.

Eigenvalues, Eigenvectors, and Determinants Math 237–005 Introduction to Linear Algebra (Spring 2019) Handout date: Tuesday March 26, 2019 Due date: Thursday April 11, 2019 Consider the matrices A= 5 3 1 9 −1 6 4 5 0 8 8 −3 1 2 6 7 b= 1 5 −3 2 . 1. Use the rref command on A − λI to check whether λ = 5 is an eigenvalue of A. 2. Use the MATLAB command c= poly(A) to find the characteristic polynomial of A. The result tells us the coefficients of the powers of λ in descending powers. Use the editor to display this polynomial. 3. Use the command polyval to determine the value of the polynomial at λ = 9. Use your answer to decide whether 9 is an eigenvalue or not. Note: use the online help polyval to see how to use polyval. 4. Find the value of the characteristic polynomial at λ = 0. Find also the determinant of A (use det). Why are the two answers the same? 5. Use the roots command on the characteristic polynomial of A to find the eigenvalues of A. 6. Use the MATLAB command [V,D]=eig(A) to find the eigenvalues and corresponding eigenvectors of A. 7. Solve the system V f = b, where V is the matrix in Problem 6 and store your solution in f . 8. Evaluate Ak b and V D k f for k = 1, 2, 4, 8 and explain what you observe and why. Submission Instructions: • The project is due at the start of class (9:30am) on the due date. • You are allowed to work in groups of up to three to complete the project. You may work individually (or in a pair) if you prefer. • Each group is required to submit both a written report presenting their solutions, and a digital appendix containing all MATLAB source code, diary/log files, and copies of any figures used saved as PNG files. • The written report can be hand-written or typeset (e.g., using a word processor or LaTex) but a hardcopy must be submitted by the deadline. A digital copy, saved as a PDF, should be included in the archive file with your MATLAB source code (see below). This copy should be a scanned copy if your group prepares a hand-written report. • The written report should include enough expository text to ensure that a reader would be able to recreate your solutions. That is, it is not acceptable to provide equations and mathematical calculations without text carefully explaining their meaning. Write your name and email ID of all group members clearly on all pages. • Use separate script and diary files for each problem. Use the naming convention Proj2type.m to label these files. For example, your script fileshould be named Proj2script.m and your diary file should be named Proj2diary.txt. Provide the names of your group members as a comment on the first line of your script for each problem. • Your MATLAB scripts should be thorougly commented so that a reader would be able to clearly understand your code. • Submit all MATLAB files (script, function, log, etc., files) and PDF of your report as a single archive file (e.g., zip file) using Blackboard before the submission deadline. Use the formula Member1LastNameFirstInitial-Member2LastNameFirstInitial-Member3LastNameFirstInitial to assign a file name to this archive with group members listed alphabetically; for example, Dr. Ames would use submit his work as AmesB.zip. Page 2 …

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