# Linear system of equations including block-matrices

Consider $Q$ which is a $NK \times NK$ symmetric positive definite ( or semidefinite) matrix, partitioned symmetrically into blocks $Q_{ij}$ which are $K \times K$ where we know that all $Q_{ii}$'s are positive semi-definite definite. Similarly, consider $S$ which is a $NM \times NM$ symmetric positive definite matrix, partitioned symmetrically into blocks $S_{ij}$ which are $M \times M$ where we know that all $S_{ij}$'s are positive definite. Now we want to find matrices $A_j$ for $j=1,2, \ldots, N$ satisfying the following linear system: $\sum_{j=1}^N Q_{ij}A_j S_{ji}= W_i$ for $i=1,...,N$, where $A_j$ and $W_i$ are both $K \times M$ matrices. Does there exist any solution for this system of equations?