Property of Moore–Penrose pseudoinverse

Question

I have a non-zero real matrix $A \in \mathbb{R}^{n \times n}$ and two diagonal real matrices $D \in \mathbb{R}^ {n\times n}$ and $K \in \mathbb{R}^ {n\times n}$. Is the following equation holding?
$$D(A)^+K = (DAK)^+$$,
where $(\cdot)^+$ is the Moore–Penrose pseudoinverse of $(\cdot)$.

Answer 1

Lets take it back to the definition of the Moore-Penrose pseudoinverse: $$DAK(DAK)^\dagger DAK = DAK$$ $$(DAK)^\dagger DAK(DAK)^\dagger = (DAK)^\dagger$$ and substitute your definition: $$DAKDA^\dagger KDAK = DAK$$ $$DA^\dagger KDAKDA^\dagger K = DA^\dagger K$$ To which it is clear that only holds if $K = D^{-1}$, as this is not given, your statement is false. The correct equation would be: $$K^{-1}A^\dagger D^{-1} = (DAK)^\dagger$$