Exercise 6.5.1

Answers

1.
Yes. See Theorem 6.18.
2.
No. Each rotation operator with a nonzero angle is a counterexample.
3.
No. A matrix is invertible if it’s unitary. But an invertible matrix, (20 0 1 ) for example, may not be unitary.
4.
Yes. It comes from the definition of unitary equivalence.
5.
No. For example, the identity matrix I is a unitary matrix but the sum I + I is not unitary.
6.
Yes. It’s because that T is unitary if and only if TT∗ = T∗T = I.
7.
No. The basis β should be an orthonormal basis. For example, we have T(a,b) = (b,a) is an orthogonal operator. But when we pick β to be
{(1,1),(1,0)}

we get that [T]β = (1 1 0 −1 ) is not orthogonal.

8.
No. Consider the matrix (11 0 1 ). Its eigenvalues are 1. But it’s not orthogonal.
9.
No. See Theorem 6.18.
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2011-06-27 00:00
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