Matrix Multiplication on Diagonal Matrices is Commutative

Theorem
Let $\mathbf A$ and $\mathbf B$ be diagonal matrices.

Then:
 * $\mathbf A \mathbf B = \mathbf B \mathbf A$

where $\mathbf A \mathbf B$ denotes (conventional) matrix product.

Proof
Note that the orders of $\mathbf A$ and $\mathbf B$ must be equal in order for matrix product to be defined.

Then we have: