# Invariance of quadratic form for unitary matrices

1. Nov 6, 2015

### spaghetti3451

1. The problem statement, all variables and given/known data

Show that all $n \times n$ unitary matrices $U$ leave invariant the quadratic form $|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}$, that is, that if $x'=Ux$, then $|x'|^{2}=|x|^{2}$.

2. Relevant equations

3. The attempt at a solution

$|x'|^{2} = (x')^{\dagger}(x') = (Ux)^{\dagger}(Ux) = x^{\dagger}U^{\dagger}Ux = x^{\dagger}x = x^{2}$.

Am I correct?

2. Nov 6, 2015

### Krylov

Yes. (Although I always need to get used to the physicist's notation for the Hermitian transpose. )

3. Nov 6, 2015

### spaghetti3451

Thanks!

I always wish I could see from the mathematician's point of view, being as I am from a Physics background.