Question: Let T' be the operator on Ro whose matrix representation is (a) Find the characteristic polynomial and minimal polynomial for T. Answer: or()) = my

Let T' be the operator on Ro whose matrix representation is (a) Find the characteristic polynomial and minimal polynomial for T. Answer: or()) = my (X) = _ (b) The eigenspace Mi associated with the smallest eigenvalue >, is the span of (1 , (c) The eigenspace M2 associated with the middle eigenvalue 12 is the span of ( (d) The eigenspace Ma associated with the largest eigenvalue As is the span of (_ (e) Find the (matrix representations of the) orthogonal projections E1, E2, and Es onto the eigenspaces M1, M2, and My, respectively. a a Answer: E1 = E2 ; Es a -a where 191 m n = _ , b = . C= , d= m = _ , and n = (f) Write T' as a linear combination of the projections found in (e). Answer: [?'] = _ E1+ E2 + - Es- (g) Find an orthogonal matrix Q (that is, a matrix such that Q' = Q 1) which diagonal- izes T'. What is the associated diagonal form A of T'

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