Homework Help: Unitary, invariant subspaces

1. May 7, 2007

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1. The problem statement, all variables and given/known data

Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that

(a) U(W) = W
(b) the orthogonal complement of W is U-invariant
(for ease of writing let the orthogonal complement of W be represented by W^.

2. Relevant equations

Unitary: U*U = UU* = I

3. The attempt at a solution

(a) first show that U(W) is contained in W, and then show that W in contained in U(W).
- U(W) is contained in W because W is U-invariant
- show W is contained in U(W)
choose x in W and show it is contained in U(W)
U(x) is in W
Is this circular thinking?
Should I instead show that the range of U (restricted to W) is W itself? Or that the nullity of U (restricted to W) is 0?

(b)
Note: U restricted to W (let’s call it U_w) is also unitary.

W^ = {x in V : <x, y>=0 for all y in W}

Now show that U(W^) is contained in W^

I’m not sure what to do now.