# Consider ℝ3with two orthonormal bases: the canonical basis

August 30, 2017

Question
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∗ ,ﬁnd 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 ﬁnd 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.

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