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## Homework Statement

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.

## Homework Equations

## 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}= U

^{H}

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 = [x

_{1}x

_{2}... x

_{n}]

Now I know that ||X||

^{2}= <X, X> = |x1|

^{2}... |xn|

^{2}

I'm not sure where to go from here. Can anyone help?