I'm really frustrated because I first thought this would be simple. Here's what I have so far.
Designate the matrix A, the eigenvalues x (since I can't TeX). Then essentially we wish to show the ||x|| = 1, meaning more specifically that x* (x-conjugate) = 1 for all eigenvalues x.
Getting there is tricky for me though. Obviously since A is unitary, the product of A with its conjugate transpose is commutatively equal to the identity. Basic algebra yields that the Inverse of A is equal to its conjugate transpose. But I have no rules about the products of matrices, esp. complex matrices to use.
I'm looking at the spectral theorem for normal operators, as the subject is in the same section, but I'm not seeing any helpful tie ins. Any help would be great.