Question

Q5 Eigenmatrix 8 Points Let C12 M2 = 211 221 : Xij ER ER} 2 22 be the vector space of 2 x 2 real matrices with entrywise addition and scalar multiplication. Consider the subspace W = {X E M2 : X = XT} of M2 consisting of symmetric matrices. (a) (2pts) Find a basis of W. What is its dimension? 1 (b) (2pts) Let A= Show that if X EW then AXAT EW. (c) (4pts) Consider the linear transformation T:W + W defined by T(X) = AXAT, where A is the same matrix from part (b). Find an eigenvector of T with strictly positive eigenvalue (i.e., a nonzero solution to the equation T(X) = \X with > 0).

Answer 1

Solution Grives W=qXG. Ma : X=XT} a let XEM be X- Xi 72 ] 23 24 x To RA Hz ra ny Hence x= XT x xa 21 Lao r3 24 x & xy & 21 and ny 9 are ng = az and arbituary.

there are 3 possibilities. Beasis in fore elements basis. When uno eg -Xzz Xyzo tyto 1 xg az -710 Xazo dz 2120 can be wel ten С» Herce mateix in ang لے [23:[::1* *1.] **[:] 11.:][:/][:)] Basis (W) Hence Dimension 3 of b het Az *** If XG W TO Show & AXAT GW ie AX AT 2 AXAT)" Since XG Ws Xe XT T Now (AXAT)T2 (AXIAT) (AT)T(Ax77 E CAB)T- BIAT] Fi (AT)*2A] AXTAT AXAT [forom 07 (AXAT) (AXAT) AXAT GW

© criven T: WW sit T(*12 AXAT since let AXATGW & T is well defined x>o be eigenalue of I T(X) for to find t to Az [! :] (ng dx some x 80 we need since let Xz 361 ng ry 1 X1 ng AX AT ng ny axy [.:\.N**11 11, AXAT Hitno - 74+Xg - txu x 4. 20g tay -Rexy su-Irgizery From x + 2xg try cout24 Аха 1.16an -ntay -20 g tay dry het za 2 + xy zdai 2 -3 & at tay xxg x tay 12 X1 - 89 + A try

p. My na deg - Subtracting from Xg [he-hell [he-he] F z lex 쉬 aya Cah Y Put un 6 Ernt dag texn z drag rar-4-d²]zo - [d2+4]xgzo since iso, 2+4 & Dha >0 rence how W= hezhe Hence Non-zero eigen vector 2-X x, 27 where 450. - [ and xi is arbituary [• :] O xi