# Normality of the product of a diagonal matrix and an SPD matrix?

I believe this to be true, but can't seem to prove it exactly: suppose $A$ is symmetric positive definite, and $D$ is a diagonal matrix. Then, $A$ is diagonal if $DA$ is normal for any diagonal matrix $D$.

Most of my noodlings have resulted in some loose conditions on $D$ and $A$ or their inverses commuting, which seems to imply that $A$ must be diagonal, but is sloppy. Is there a more direct proof?