1. Consider R3with two orthonormal bases: the canonical basis e=(e1,e2,e3) and the basis f=(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 map T∈?(R3) with eigenvectors f1,f2,f3 and eigenvalues 1, 1/2, -1/2, respectively.
2. For the following matrices, verify that A is Hermitian by showing that A=A∗ ,?nd a unitary matrix U such that U-1AU is a diagonal matrix, and compute exp(A).
A = 5 0 0
0 -1 -1 + i
0 -1 - i 0
3. For the following matrices, either ?nd a matrix P (not necessarily unitary) such that P-1AP is 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 ) are positive operators on V . Prove that S + T is also a positive operator on T.