Thread: 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?

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 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.

Tonio


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

.