Matrix equation solving

Question

My question is very simple, but I it seems to be impossible for me to figure it out. Here we go, find $\mathbf Z$ such that $$ (\mathbf Z^t\mathbf Z)^{-1}\mathbf Z^t\mathbf W=X_1Y_1+X_2Y_2 $$ Where $\mathbf Z$, $\mathbf W$, $X_1$, $X_2$, $Y_1$ and $Y_2$ are all matrices.

Answer 1

If we assume that such a $Z$ exists with linearly independent columns, then $( Z^t Z)^{-1} Z^t = Z^+$ is the Moore-Penrose pseudo inverse of $Z$. So you could try to solve $ U W = X_1Y_1+X_2Y_2$ first by vectorization and afterwards recover $Z = (Z^+)^+ = U^+$.