1. Consider3with two orthonormal bases: the canonical basise=(e1,e2,e3)and the basisf=(f1,f2,f3), where

f1=(1,1,1)/(3^(1/2)) ,f2=(1,2,1)/(6^(1/2)) ,f3=(1,0,1)/(2^(1/2))

Find the canonical matrix, A, of the linear mapT(3)with eigenvectorsf1,f2,f3and eigenvalues1,1/2,1/2, respectively.

2. For the following matrices, verify thatAis Hermitian by showing thatA=A,nd a unitary matrixUsuch thatU1AUis a diagonal matrix, and computeexp(A).

A = 5 0 0

0 1 1 + i

0 1 i 0

3. For the following matrices, either nd a matrixP(not necessarily unitary) such thatP1APis a diagonal matrix, or show why no such matrix exists.

A =5 0 0

1 5 0

0 1 5

4.Let V be a finite-dimensional vector space over F, and suppose that S, T L(V ) arepositive operators on V . Prove that S + T is also a positive operator on T.