I have a positive-semidefinite matrix $A$, and a positive diagonal matrix $D$. I want to find the Moore-Penrose pseudoinverse $(DA)^+$.
Is there any formula for $(DA)^+$ in terms of $A^+$ and simple operations on $D$? If it helps, I have full knowledge of the kernel of $A$.
$(DA)^+ \stackrel{?}{=} A^+ D^{-1}$ is tempting but, unfortunately, wrong. Of course I can do a generalized spectral decomposition
$$AV = D^{-1}V\Lambda$$ from which $(DA)^+ = V\Lambda^+ V^{-1}$, but $V$ and $\Lambda$ depend on $D$ in a rather complicated way.