Question

Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D such that D = P-1AP. f. (2 pts) Find two other matrices P4 and P2 and corresponding diagonal matrices D, and Dą such that D1 = P11 AP, and D2 = P7 AP2.

Answer 1

A=11 o on 6 3 2 1 Lag -1 o. a) characteristic polynomial fart, a 2o PA (2) det. (A-AI) a deet ( 6 32 2. az 1 2 (1-2) (3-1) E-x) = 2x-2x (1-2) = (1-2) [-37 ta²+2]. = (1-2) [2²-3x+2] (1-x) [x²21-9 +2] = -(1-2)*(1-2) 6) eigenvalues are BAQ) 20 722 1, 2. c) for any eigen value a of A, ] I such that Av=au A2I) U=0,

0. tar a=1. roo 07137 16 2 29 3 -1 -1[2] loj = GXt Q4+2 z=0.; -32-34-82=0. =. 32+y+z=0; 3x+y +Z. 20. dimension of eigen space corresponding to aəlin 2. for a=2 - 1007 [-3-1 -2] -> 2=0; 6x+y +2220) -3x-y-2Z20. => x=0; y+2220 ; y+2220 =).2=0, 9+2220 dimension of eigenspace corresponding to 222 i 1. d) since far every eigenvalue of A we have gon geometrie multiplicity (= dimension of eigenspace) equal to algebraie multiplicity, then, Ais diagonalizable.

e) Bars of E2 = {(0, 2, 1)} & Basis of Eg > { (1, 2, 4), (2, -4,2)} P2 1 1 0 1-4 -2 D 0 1 0 1) Piz 1 P, TO 1 170 D, 2 1 2 0 07 1-2 1 -4 o 1 o -4 Lo P2 251 o 17. 1 1 -2 4 Delo o7 1 -2 4 2 0 2 0 L-4 1
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