Unitary matrices are invertible.

Question

I'm currently looking for the proof that a matrix $Q \in \mathbb{K}^{n\times n}$ that is unitary (i.e such that $QQ^*=I)$ is invertible and that $Q^* = Q^{-1}$.

For square matrices, if $AB=I$, then $BA=I$. – user1551 Mar 08 '18 at 18:40 — user1551, Mar 08 '18 at 18:40

lisyarus · Accepted Answer · 2018-03-08T19:18:03.083

2

A square matrix $A$ is called invertible if there exists $B$ such that $AB=I$, the identity matrix.

Compare this with the equation $QQ^*=I$ and conclude.

edited Mar 08 '18 at 19:18

answered Mar 08 '18 at 18:38

lisyarus

15,517

score 0 · Answer 2 · answered Mar 08 '18 at 18:38

0

The condition that $Q^* Q= QQ^*= I$ by definition means that this is invertible, with inverse given by $Q^*$.

answered Mar 08 '18 at 18:38

DKS

1,280

Unitary matrices are invertible.

2 Answers2