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.

$\ds \mathbf A$

$:=$

$\ds \sqbrk {a_{ij} }_n$

$\ds \mathbf B$

$:=$

$\ds \sqbrk {b_{ij} }_n$

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

Then we have:

$\ds \mathbf A \mathbf B$

$=$

$\ds \sqbrk {c_{ij} }_n$

$\ds $

$=$

$\ds \begin {cases} a_{i i} b_{i i} & i = j \\ 0 & i \ne j \end {cases}$

$\ds $

$=$

$\ds \begin {cases} b_{i i} a_{i i} & i = j \\ 0 & i \ne j \end {cases}$

$\ds $

$=$

$\ds \mathbf B \mathbf A$

$\blacksquare$