I have a problem of the form
\begin{equation}
\mathbf{A}\mathbf{\Gamma}_1\mathbf{A}^\top\mathbf{A}\mathbf{\Gamma}_2\mathbf{A}^\top=\mathbf{C}
\end{equation}
where $\mathbf{\Gamma}_1$ and $\mathbf{\Gamma}_2$ are diagonal and $\mathbf{A}$ is nonsquare. I would like to solve for the product $\mathbf{\Gamma}_1\mathbf{\Gamma}_2$ by minimizing an expression
\begin{equation}
\min_\mathbf{X} ~ \left\| \mathbf{E} \mathbf{x} – \mathbf{c} \right\|^2
\end{equation}
where $\mathbf{x}=\text{vec}\left(\mathbf{\Gamma}_1\mathbf{\Gamma}_2\right)$ and $\mathbf{c}=\text{vec}(\mathbf{C}$). I know that for the simpler expression
\begin{equation}
\mathbf{A}\mathbf{\Gamma}\mathbf{A}^\top=\mathbf{C}
\end{equation}
it's possible to use the Kronecker product to obtain
\begin{equation}
(\mathbf{A}\otimes\mathbf{A})\text{vec}(\mathbf{\Gamma})=\text{vec}(\mathbf{C})
\end{equation}
which does the trick in that case. Is there anything analogous that would work for my problem?
Best Answer
Define the matrices $$\eqalign{ B=A^TA,\quad F=A^+C(A^+)^T,\quad X=\Gamma_1,\quad Y=\Gamma_2 \cr }$$ where $A^+$ denotes the pseudoinverse of $A$.
Let's also use a naming convention where uppercase letters are matrices and the corresponding lowercase letter is the vector formed from the main diagonal, e.g. $$\eqalign{ x = {\rm diag}(X),\quad y = {\rm diag}(Y),\quad f = {\rm diag}(F),\quad etc }$$ Solve the given equation for $(XBY)$ in a least-squares sense. $$\eqalign{ AXBYA^T &= C \quad&\implies\; XBY = F \cr }$$ Then extract the diagonals and isolate the term of interest. $$\eqalign{ {\rm diag}(XBY) &= {\rm diag}(F) \cr x\odot b\odot y &= f \cr x\odot y &= f\oslash b \;= {\rm diag}(XY) \cr \Gamma_1\Gamma_2 \;=\; XY &= {\rm diag}^{-1}(f\oslash b) \cr }$$ where $(\odot, \oslash)$ represent elementwise/Hadamard multiplication and division, and the ${\rm diag}^{-1}(v)$ operator creates a diagonal matrix from the input vector.
NB: The formula to rewrite $\,{\rm diag}\big(XBY\big)\,$ as the Hadamard product of vectors does not apply to general matrices; it is only valid when $X$ and $Y$ are both diagonal matrices.
Because $B=A^TA$, none of the elements on the diagonal (i.e. the vector $b$) will be zero unless $A$ contains an entire column of zeros. Therefore it should be safe to perform the indicated Hadamard division $(f\oslash b)$.