# Unitary Matrices

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• Mar 29th 2010, 02:20 PM
apple123
Unitary Matrices
The question:
Show that ||UX|| = ||X||.

My attempt

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?
• Mar 29th 2010, 03:28 PM
tonio
Quote:

Originally Posted by apple123
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?

.