Unitary Matrices

Definition: Unitary Matrix

A unitary matrix is a complex square matrix $U\in {\mathbb{C}}^{n\times n}$ whose inverse is its Hermitian transpose.

$$U^{-1}=U^{†}$$

Theorem: Orthonormal Bases from a Unitary Matrix

If $U\in {\mathbb{C}}^{n\times n}$ is a Unitary Matrices, then:

• the columns of $U$ form an Inner Product Spaces of the vector space ${\mathbb{C}}^n$;

• the columns of its Hermitian transpose $U^{†}$ also form an Inner Product Spaces of the vector space ${\mathbb{C}}^n$.

PROOF

TODO