Math 110, Fall 2015. Homework 11, due Nov 13. Prob 1. Let T be a self-adjoint Operator on a nite-dimensional inner product Space (real or complex) such that A1, A2 E ]R are the only eigenvalues of T. Prove that p(T) = 0 where p(/\\) :=()\\_/\\1)(2\\ A2). Give a counterexample to this statement for an operator which is not self-adjoint. Prob 2. Let T be a normal operator on a complex nite-dimensional inner product space V whose distinct eigenvalues are A1, .. . , A}: E C. For any n E V such that \"v\" = 1, show that k: (T1), 'U) = Zajhj 3'21 for some nonnegative numbers (15;, j = 1, . . . , k, that sum up to 1. Prob 3. Let T E (XV). Show that (v, u) :=(T'v,u) is an inner product on V if and only if T is positive (per our denition of positivity). Prob 4. We already know (how?) that the operator T = D2 is nonnegative on the space V := Span(1, cos 3, sin z) over IR, with the inner product (f, g> := :f(z)g(m)dm- Find (a) its square root operator x/T; (b) an example of a self-adjoint operator R 75 x/T such that R2 = T; ( c) an example of a non-self-adjoint Operator S such that 3*.5' = T. Prob 5. Let T1 and T2 be normal operators on an ndimensional inner product space V. Suppose both have n distinct eigenvalues A1, . . . ,AH. Show that there is an isometry 5' E (V) such that T1 = S*T2.S'. Prob 6. Find the singular values of the Operator T E p2(C) : p($) I> rp'($) + 2:02;)\"(3) if the inner product on 732(0) is dened as 1 (m) == / maa. 1