I'm new to stackexchange so feel free to correct my style/format/logic etc.
The question is this: let's say $A$ is a square matrix of size $n$. I would like to show that $AD = DA$, for any diagonal matrix $D$ also of size $n$, if and only if $A$ is also diagonal.
I think I have some of the proof but am not very confident in it.
$\Leftarrow $: If A is diagonal, it is not too hard to show that $AD = DA$, because multiplying two diagonal matrices just amounts to multiplying the corresponding diagonal entries.
$\Rightarrow$: (i) $DA$ is found by multiplying each row in A with the corresponding entry along the diagonal in $D$. $AD$ is found by multiplying each column in $A$ with the corresponding entry along the diagonal in $D$. Since $AD = DA$, this product has to be symmetric.
(ii) Now suppose $A$ weren't a diagonal matrix. Then if we make the entries along the diagonal in $D$ all different, $AD$ won't be symmetric anymore (?). This contradicts (i), so we have shown both ways.
Does this work?
[edited]
