Question

3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V → V be the linear operator defined by 0 T(1) = ( ; ;) A. (i) Compute To((.) (ii) Determine whether or not there is an orthonormal basis of eigenvectors B for which [T]is diagonal. If such a basis exists, find one.

Answer 1

Q:3. consider the space V = M2xiz (€) arid let To Vybe the linear op operator defined by (iOA in Compute T i T(A) T* 0 (0 :)) () (ME : ) fi ( )( ) 0 ) TT riti 0 0 0 -ity if (ii) In short we need to check T is diagonalizable We compute charac teristic polynomial for t. det (ET Ja – 11) = 2+1E0 o-a ci 0-2 a= I So we have distinct eigenvalues of I so Tis diagonalizab. and hence a basis of eigenuectors exists, lets compute the bans. (or elgennectors),
it
ev is an Eigenrector for 2, = i, If x= ( 2 1 2 1 zle for. A = i then T(X) = 1, X. =) 13:27 ) eigennect. ) - Cene iz;) 2 Zz 1 Z₃ Z4 'iz, izz liz, iza iZz izu izz izi izz . (iz izz izu 21 = Z₂ & 2₂= 24. x o > (22) is an eigen vector for Aici where 2,3 Zz e €¢. Eigennector for A2=-1, similarly, izz izu -iz, -iz =) 2,=-Z, 8223-24 - 1Z3 - izu ** (1 ev is an eigennector for Apari with ( o 2 Zz 2₃ 24 ) ; frizi -izz -iz, -iza H li رک iz, izz 10 x 2. Zz - Zi -Zz o 21,22€¢.

Now to form a basis for V, we fix 2, 372.68. het zizi, Zz =-1. Then i i ez Piz (-); e (. :) So B = { el, ez) is a basis for V-M2x2 (Ⓡ), weonly need to check if it is orthonormal? Pilz = tor la MB) = toile prez) ri ож 1. P23 (*:)*)-(*)-(3) for lett. 12) = 6 hence (e1, P2) so they are orthogonal. To check orthonormality, see for <e,,?,)&& (92, P2} & should be 1. Lene z 02:1-6 (1+1 -1-1 1-1-1 1+1 1:0) - 63 -2) So {, ei?p=4. (HI H HI HI So <e2, 12) = 4. 2. We divide e, der by 4 to get enthonormal weeters forming a basis for v = M272 C¢).