Linear system of equations including full rank matrices

Question

1- Consider a matrix equation as $Q_{m \times m}X_{m \times n} S_{n \times n} = Z_{m \times n}$ where we know that $Q$ and $S$ are full rank matrices. Then, we know that the solution is $X = Q^{-1}ZS^{-1}$.

2- Now consider the following system of equations:

$$ \sum_{j=1}^N Q_{m \times m}^{ij}X_{m \times n}^j S_{n \times n}^{ji} = Z_{m \times n}^i \quad i=1,2,\ldots, N. $$ where $Q^{ij}$ and $S^{ji}$, $i,j=1,2,\ldots, N$ are full rank matrices.

Does there exist any $X^j$ matrices for $j=1,2,\ldots, N$ satisfying this equation (at least one)? It seems that problem 2 is similar to problem 1, but I don't know how to solve it. Any idea?

Answer 1

Matrix equation $Q_{m \times m}X_{m \times n} S_{n \times n} = Z_{m \times n}$ can be written as $(S^T \otimes Q)vec(X) = vec(Z)$ where $vec(X)$ denotes the vectorization of the matrix $X$ formed by stacking the columns of $X$ into a single column vector and $\otimes$ is Kronecker product. Using this, Problem 2 can be written as, $$ \sum_{j=1}^N ((S^{ji})^T \otimes Q^{ij})vec(X^j) = vec(Z^i)\quad i=1,2,\ldots, N. $$ Let define $H^{ij}=(S^{ji})^T \otimes Q^{ij}$, $Y^j = vec(X^j)$ and $W^i = vec(Z^i)$. We can show that $H^{ij}$'s are full rank matrices. Then $$ \sum_{j=1}^N H^{ij} Y^j= W^i\quad i=1,2,\ldots, N. $$ which can be written as $H Y = W$ where $$ Y = \begin{bmatrix} Y^1\\Y^2 \\ \vdots \\Y^N \end{bmatrix} \quad W = \begin{bmatrix} W^1\\W^2 \\ \vdots \\W^N \end{bmatrix} \quad H =\begin{bmatrix} H^{11} &\ldots & H^{1N}\\ H^{21} &\ldots & H^{2N} \\ \vdots &\vdots & \vdots\\ H^{N1} &\ldots & H^{NN} \end{bmatrix} $$

All solutions of $H Y = W$ (if any exist) are given as $Y = H^{\dagger} W + (I - H^{\dagger} H) T,$ where $T$ is any arbitrary vector with the same size as vector $W$.