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.
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 is unitary if , and then
we have and we're done
Say X = [x1 x2 ... xn]
Now I know that ||X||2 = <X, X> = |x1|2 ... |xn|2
This should be , and not a product.
I'm not sure where to go from here. Can anyone help?