__The question:__
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.

__My attempt__
I'm not sure where to start. So I looked up the definition of a unitary matrix.

You had to? I mean, if you're asked to do this exercise then one could expect you know at least what a unitary matrix is
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

Who knows what you meant by the first condition. Anyway, a matrix $\displaystyle U$ is unitary if $\displaystyle U^{-1}=U^{*}=\overline{U^t}\Longleftrightarrow U^{*}U=UU^{*}=I$ , and then we have $\displaystyle \|x\|^2=<x,x>=<x,U^{*}Ux>=<Ux,Ux>=\|Ux\|^2$ and we're done
Say X = [x1 x2 ... xn]

Now I know that ||X||2 = <X, X> = |x1|2 ... |xn|2

This should be $\displaystyle x=(x_1,\ldots,x_n)\Longrightarrow \|x\|^2=\sum^n_{i=1}|x_i|^2$ , and not a product. Tonio
I'm not sure where to go from here. Can anyone help?