When would be the solution $C$ that minimizes $\lVert A-ZC\rVert$ be the same as the one that minimizes $\lVert Z^{-1}A - C\rVert$ or say some scaled version $\alpha C$? All entries in square matrices $Z$, $A$ and $C$ are real.
Under what conditions is it invariant upto an orthogonal transformation say $P + qDC$ where $P$ is arbitrary matrix, $q$ is arbitrary nonzero number and $D$ is an arbitrary orthogonal matrix.
I tried equating the two solutions to get the condition $$\left(Z^{2}\right)^{-1}Z^2 = \left(M^TM\right)^{-1}M^T$$ where $M = P + qDC $, if $Z$ is symmetric, and if not the condition is $\left(Z^TZ\right)^{-1}Z^T = \left(M^TM\right)^{-1}M^TZ^{-1}$. I ask this question to see if there is a better or broader take on this. In addition, would there be more to add if $C$ was always skew-symmetric and $Z$ was symmetric?