Applied Mathematics 1201A/B Chapter 2.6: CH 2.6 Eigenvalues and Eigenvectors

4. Let 1 0-1 2 3 1 A--1 3 02 2 2 2 2 5 -2 3 0 2 -2 6 (a) [10 marks] Define and compute the I1 operator norm of A. (b) [7 marks] Prove that any eigenvalues of A are real. (c) [8 marks] Prove that any eigenvectors of A corresponding to different eigenvalues are orthogonal to each other (in the standard, Euclidean inner product) Hint: (b) and (c) can be answered without computing explicitly the eigenvalues/eigenvec- tors of A. In fact, you only need to use one property of A, which can be easily checked. Total: 25 marks

Given the matrix B = [3 8 -4 0 -1 0 2 4 -3]. (i) Find all the eigenvalues of B. (ii) Find the corresponding eigenvectors of B. (iii) Diagonalize matrix B. (iv) Using the results of (i)-(iii) find B^2013. (b) Consider the following Linear Programming problem: Find values of x and y which maximize z = 5x + 6y subject to constraints: x + y lessthanorequalto 6, 2x + y lessthanorequalto 10, 3x + y greaterthanorequalto 3, where x, y are non-negative numbers. (i) Sketch the set of feasible solutions. (ii) Find the extreme points of the set of feasible solutions. (iii) Find the optimal solution.

Research Project 3 MATH 334 - MODERN ALGEBRA In this research project, you will be studying a group not studied in class. will be to first show that the set described actually is a group and then to properties of this group. This research project will consist of 2 parts: Your e some 1. A written report which must be typed. You should give complete solutions and explanations, written in full sentences, to all parts of your research questions in this portion. This portion determines 75% of your grade. 2. A powerpoint or other computer software presentation. You will need to present your findings to the class in a 10-15 minute presentation. You do not need to give full solutions to all aspects, but you should include the most important/most interesting things you found. This portion determines 25% of your grade. Consider the following operation on R. 1. Show that this operation makes R3 into a group. This means that you need to show the following: (a) Multiplying two elements using this operation produces an element of R3 (b) The operation is associative. (c) There is an identity element (what is it?) (d) Every element has an inverse. 2. Is this group Abelian or nonabelian? 3. Consider the set of matrices of the form 01 b where a,b and c are real 0 0 1 numbers. Show that the above group is isomorphic to the set of these matrices under matrix multiplication. 4. Show the set of matrices of the above form but with a, b, and c restricted to being 5. What is the determinant of each matrix in this set? 6. What are the eigenvalues and eigenvectors of the matrices in this set? 7. Show that the set of 3 x 3 upper triangular matrices with nonzero entries on the integers is a subgroup. diagonal forms a group and that the above group of matrices is a subgroup of this group. Is it a normal subgroup?