**Theorem: ** The product of two unitary matrices is unitary.

**Proof:** Let and be unitary, i.e.,
and
, then
is unitary:

**Theorem:** Two square matrices and are simultaneously
diagonalizable if and only if they commute.

**Proof**
(reference)

- Let and be simultageously diagonalizable by

then

- Let and commute, i.e.,
.
Assuming is an eigenvector of corresponding to eigenvalue
, i.e.,
, then

we see that is also an eigenvector of corresponding to the same eigenvalue , i.e., must be a scaled version of (in the same 1-D space): , i.e., is also an eigenvector of .

**Theorem:** A matrix is normal if and only if it is unitarily diagonalizable.

**Proof**
(reference)

- If is unitarily diagonalizable:

where is unitary and is a diagonal matrix satisfying , then is normal:

- If is normal, then it is diagonalizable by a unitary matrix.
First we show any matrix can be written as

where

are both Hermitian, and diagonalizable by a unitary matrix. As is normal, we have

We see that , i.e., and commute, and they can be simultaneously diagonalized by some unitary matrix :

and so can :