Question

Let A= 1. Without computing anything, can it be true that A is diagonalizable? What can you say about its eigenvectors? 2. Find the eigenvalues of A and one eigenvector of A. 4. Use the spectral theorem to find another eigenvector of A. 5. Find an orthonormal eigenbasis for the transformation defined by A. 6. Find orthogonal S such that s-1 AS is diagonal. 7. Is there S such that ST AS is diagonal? 8. Prove or disprove that there is a unit square in R2 such that A takes that square to a rectangle whose sides are length 2 and 4

Answer 1

R 2) - [?] 3 The given matsix Ais Symmettic matsix.ie. A=A? We know symmetric matolesore alwalys diogenizable & have have eigennectors which are ofthogonal. These exists an orthosmal basis of eigennectors for A. .; His dio y on alizable and have orthogonal eigencedors. The eigenectors of A form an orthonomas set spanning A 3 3 • Eigenialues: The characteristic equation is given by IA-d Il = 0 1 - ?x=0 (Ind) (red)-q=0 CX 24-8)= 0 (1+2)(1-4)=0 1+2=0 og dohoo x = -2,h d=-2 and dan Eigenvalues of A are: Eigenvector For eigenvalue d=-2, A-11: [] +2 [53] +2 [0 ] + [5}]+[..] = [3] Now, reduce the matrix, R,KR, 3 [';] 3 3 R24 R₂-3R, [oi] :. 3

The system associated with the eigenvalue d=12, A +21) CA (0) 1 X, +2,=0 =) X, F-X2 .? Eigenwetor coresponding to the eigenvalue 12-2 is [- VE 22 x2 2 let X₂=1 11 vi [] 4 ) Using speetal theorem, we al deady know that matrix A hag an Orthomomal eigen hagis since it symmetric matrix. Thus, we have to find eigen space for the other eigenvalue (h) is ofthogonal to this one. [:] For eigen calle Ach, u, ie. Nga E

Checking :. Both are ofth normal 5) [7].[:] = -1-1=0 C:] Hese, lu Uz E using Orthomalization, vi a Vi lu, il VEO?4 (112 1 [1]** [:] VIDA va [i] - Como [:] 빌 - Uz VORCU? Vz [ llu 'Yuz :. Ostho normal Eigenha sis is given by, B= (vi, v₂ E 6) The s matrix, Sa [vi vz] ar fi k to

SAS: 3 [ sl- Slast Tiliter :] ho Here, A- si Sle 3 re Yra Yr aire le Yuz Yuz 3 r2 • 12 [ I va 202 22 5 AS a =D เร เร [3] :,]+[3] Thus, the required s mathi'x is, De o Sa Tus iž to Liz te 7) Yes, there is s such that Stasis a diagonal Since, 5-1=gt ie s= 8) yes, the square with sides of lengths one and herticos Hva 2102 .-lYu2 o Any squase with whit edges two edges along the lines y=x x Ç your will wolk