Question

6. Matrix Calculations: Consider the linear tre tions: Consider the linear transformation represented by the matrix (a) {5 points) Compute the eigenvalues of A. (b) {10 points) Compute the eigenvectors of A. 1 points is the state space system = Ag internally anymptotically stabler Explain.

Answer 1

- G. Alu-27 2 @ eigen values of A characterstic equation - x = eigen value IA-XI1=0 i on 16 11.78:1 - glo il 5. dentity model na A :AD-BC : 1-2 171 de terminant ( -»)(1-2) – (-2)(-2) =0 4-67-7+7"-450 44-5120 X(X-5)=0 2:0 6 а ези - eigen values of A are 0,5 (b) eigen vector of A to find eigen vector use (A-71] (x1=0 - X= eigen vector

CA-][X1-0 pernah 167:19] +7=0 eigen value R2726, +2 R1 11*1687 uni-282=0 21-22=0

let xat 24-02=0 need to consider M2=2+ x=6512=647 ++ [1] t is just a scalar no if eigen value = o then eigen rector = 12 =) eigen value =5 [A25][x 1-0 14.0701 21:17 225 T 110 37-187 R₂ - R2-28 176 71867:18] -N, -222=0 let xi=t at = 2x2 al X2 = -t

*(5,:1.837:+14) - if eigen value = 5 then eigen rectoral - (C) giver g=A9 a we know characterstic equation 1A-2910 -) similarly if we take in state space system then charactastic equation is / SI-Al=0 - both are equal only but just "a is replaced - with "S" (.€ = 155-Al=0 - (5-4)25-10-450 3-5544-450 i 5-55=0

s(5-5)=0 s=o s=5 to * one pole lie at jw anis 15.0] * one polke lie at right hand side [s=5] of it poles be at right hand side system unstable ... The state space system q = Aq is internally asymptotically unstable. is