Unitary Matrices as a Group: Proof and Properties

[spaghetti3451]

Homework Statement

Show that the set of all $n \times n$ unitary matrices forms a group.

Homework Equations

The Attempt at a Solution

For two unitary matrices $U_{1}$ and $U_{2}$, $x'^{2} = x'^{\dagger}x' = (U_{1}U_{2}x)^{\dagger}(U_{1}U_{2}x) = x^{\dagger}U_{2}^{\dagger}U_{1}^{\dagger}U_{1}U_{2}x = x^{\dagger}U_{2}^{\dagger}U_{2}x = x^{\dagger}x = x^{2}.$

So, closure is obeyed.

Matrix multiplication is associative.

The identity element is the identity matrix.

$x'^{2} = (U^{-1}x)^{\dagger}(U^{-1}x) = x^{\dagger}(U^{-1})^{\dagger}U^{-1}x = x^{\dagger}(U^{\dagger})^{-1}U^{-1}x = x^{\dagger}(UU^{\dagger})^{-1}x = x^{\dagger}x = x^{2}$.

So, the inverse of any unitary matrix is a unitary matrix.

Is my answer correct?

 

[fresh_42]

yep, and if you mention that the identity is clearly unitary and the inverse exists at all, e.g. because |det(U)| = 1 it'll be perfect

 

[spaghetti3451]

Thanks! Got it!