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Let (V, C) be a finite-dimensional complex inner product space. We recall that a map T...

Let (V, C) be a finite-dimensional complex inner product space.

We recall that a map T : V → V is said to be normal if T∗ ◦ T = T ◦ T∗ .

1. Show that if T is normal, then |T∗(v)| = |T(v)| for all vectors v ∈ V.

2. Let T be normal. Show that if v is an eigenvector of T relative to the eigenvalue λ, then it is also an eigenvector of T∗ relative to the eigenvalue λ.

3. When is the normal operator T self-adjoint? Motivate your answer!

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