Math1853 Homework 2 (Released: 11 Oct 2019 Due: 20 Oct 2019 11:59PM) Questions 0 1. (15%) Find all eigenvectors and eigenvalues of the matrix 2 13 and comment if it is diagonalizable or not. 2. (10%) Find the determinant of M = 3. (15%) Assume A E R4X4 has eigenvalues 1, 2, 3, 4. Find (52 + (A - 51|) 999 where | o | denotes determinant. (Hint: Note A has distinct eigenvalues and can be put into an eigendecomposed form A = VAV- such that [A - 51| = [VAV-1 -5VIV-1 1.) 4. (a) (10%) Assume matrix A has two distinct eigenvalues 1 and 12, their corresponding eigenvectors are v1 and v2, respectively. Show v1 + 12 is not an eigenvector of A. (b) (5%) If A E Rnxn is invertible and diagonalizable, prove that A-1 is also diagonalizable. all a12 a13 a14 5. A = a 22 a 23 a 24 0 0 a33 a34 is an upper triangle matrix. Prove the O 0 0 a44 statements below: (i) (5%) If the diagonal entries are distinct, then A is diagonalizable. (ii) (15%) If a12 = a34 = 0, and all = a22 = 1, a33 = 044 = -1, then A is diagonalizable. 6. (15%) Assume v is an n x 1 column vector, I is the n x n identity matrix. Prove that A = I - ,7, vv is an orthonormal matrix. 7. (10%) Assume W E Rnxn is an invertible square matrix, and A = WTW and x E R", prove that f = x A x is always positive. 1