MATH 225 Spring 2017: Assignment 9 Due at 8:25 a.m. on Friday, July 21, 2017 It is important that you read the assignment submission instructions and suggestions available on LEARN. 3 0 0 1 2 0 1 1 is a 1. A real canonical form of a 3 3 real matrix A is B = 0 0 1 and P = 0 0 1 0 2 2 0 1 change of coordinates matrix such that P AP = B. (a) What are the eigenvalues of A? (b) For each eigenvalue of A give a corresponding eigenvector. Page 1 1+i 1i 2. Let ~u = 1 and ~v = 2i . Using the standard inner product for C3 , calculate the following: 3i 1 (a) h~u, ~v i (b) ||~u|| (c) h2i~u, ~v i (d) h~u, 2i~v i Page 2 6 0 4 1 . The characteristic polynomial of A is p() = 3 + 22 4 + 8. Working 3. Let A = 0 1 8 1 5 over C find a matrix P and a diagonal matrix D such that P 1 AP = D. Page 3 4. For matrix A given in question 3, determine a real canonical form B and give a change of coordinates matrix Q such that Q1 AQ = B. Page 4 \u0012\u0014 \u0015\u0013 \u0014 \u0015 x y 5. Let L : be the linear operator defined by L = . Prove that the only real y x invariant subspaces of L are {~0} and R2 . R2 R2 Page 5