Question

0 2 0 A = 20 Lo 0 1 then the eigenvalues of A are 1 = 1,2= 2 and X = -2 (all with multiplicity of 1) (A) Find an ordignormal basis for the eigenspace corresponding to X=1 (B) Find an onHonormal basis for the eigenspace corresponding to 1 = 2. (C) Find an arxhonormal basis for the eigenspace corresponding to 1 = -2. (D) Provide an orthogonal matrix P and a diagonal matrix D such that PAP=D.

Answer 1

solution: Given matrix Given that the eigenvalue of A are di=1, 22=2 and x = -2 all eigenvalue multiplicity a's one. > A) Compute the orthonorincel basis for the eigenspace corresponeu'ng to dial Now first fing the eigenspace of corresponding eigen veice doar bet consider the system deten til de CA-dil v = 0 where v is any vector v= [X,,He, x3] s is eigenvalue of A, I is identity of matix of climenzion a Po-,20] [3,7 2 0-2,0 X220 O -RJ (23_ BroeIn 2 0727 2 -1 0 *2 lo lo o o] Now solve the homogenous system of : linear equation by using Gauss elimination method. Gaussian

consider augmenteel machix | 2 -1 010 R =) (-DR, fi z olo 2 - 0 Loo olol R2-> R2-2R, - zo 1/2 R2 = -2 R1 R172 R2 r oolol - Write the dinear equcetion system, zly 30 x2 =o .: Equchon i's two and variable iš 3 1 then one variable is free variable I let Xz-t a solution is ro] ro
a
.: Eigenspace of corresponding eigen velue is V, sro Vi is orthogonal vector Now find the orthonormal basis of V is to divideel the vector it's Orthonormeel basis norm. Villa Vosto 2: = 5 Orthonormeel basis of Visſ B ] da= 2 Now first find the eigen space of corresponding eigen value let 12 = (x, H2, H3) be any vector Consider the system CA-d2I) V2 = 0 Toode 2o Te | 2 odzo To o Hz 212 - 2 on 12 -2 o Lo 0 - L To solve the homogenous system of eeju celion by using Geussian elimincehon methoch, EN9

i consider augmenteel mutia F-2 201 | 2 -2 o t Rial yol 12 -2 ol 7 Rz7R2-2Ry ſi - ó og - R26R3li - Oro 1 Loo olol Write in linear Pelucehon system 2,-*2=0 Sep 272 3 -0 : - One Warliceble is free variable dzão let x2zt is free variable. then wist solution is V2 = x 1 72 ] [e] t 1 = { ... The eigenspace of corresponeling eigenvalue o2 = 2 é's Vzo V2 is orthogonal rector Now find the Norm otva livello ST2+20 = IIHTO = 12 ...Ortho normal basis of v2 = (fe, Šio).

3 l3= -2 Now find the eigenspace of eigenvalue dy -2 Consider the system (A-MI) V₂ = 0 where V3 t [X1, #2, #3) be uny veetor Ford3 2o Tal lo 1xz 2 Lo ondo o 14 12 2 0 2 2 0 L 0 33l To solve the homogenous system by using Gaussian eliminidyon method. Consicier augmented matrix [ 22olor 12 2 o o wo R2J Roers / o Ri+ [1 0 2 2 0 Joo 3 0 o o loo o R2 R3 l lol Lo o öld Write in linear ecuation system antol2 20 = X1=-X2 3x3=0 23-0

X₂ is free variable, let x2 = { then x=-4 Solution is ſ mes 1 X2 : Bh V3 is eigenspace of corresponding eigenvekte d3a-2 Orthonormal basis of V3 Wall S-13241 ²to2 = V1212 to 52 Orthonormal basis of V3 =

D J Now find the orthogonal matrix p The eigenvalues are distinct the eigenveetor are orthogoneel to one another, once we now dinide each of them by it's length we obtain the coloums of the orthogonal matrix Up =(0,0,1) =(0,0,1) V2=(1,4,0) =G Š , ) V2 = 64,1,0) - ) pa po 1537 This orthogonal matrix of matrix a р. Го о ) 152 152 0 |- 172 112 of such theet PTA p = D. Consider prap - Tool 70 207 Toys 1/5 -11527 1152!(52 0 2 0 0 0 1152 115 L-'15? Iko lo 0 D o 0] : Co o il po 15 -1/17 2/52 2/5 ollo 1152 1152 (2152 -2/12 o li 0 0

| lo2 01 lo -2 PTAP =D Hence the requricel solution