U is a unitary matrix. Show that
||UX|| = ||X|| for all X in the complex set.
Also show that |λ| = 1 for every eigenvalue λ of U.
The Attempt at a Solution
I'm not sure where to start. So I looked up the definition of a unitary matrix. It satisfies one of these conditions:
U-1 = UH
The rows of U are an orthonormal set in the complex set
The columns of U are an orthonormal set in the complex set
Say X = [x1 x2 ... xn]
Now I know that ||X||2 = <X, X> = |x1|2 ... |xn|2
I'm not sure where to go from here. Can anyone help?