Linear Matrix Equations Controllability and Observability, and the Rank of Solutions
Hȧrald K. Wimmer · SIAM Journal on Matrix Analysis and Applications · 1988
The equation \[ (*) \qquad \sum_{i,k} f_{ik} A^i XB^k = C \] is studied. The controllability matrix of $( A,C )$ and the observability matrix of $( B,C )$ yield bounds for the rank of X. If the solution X is unique it can be expressed in the form \[ X = \sum_{i,k} h_{ik} A^i CB^k . \] The coefficients $h_{ik} $ are determined by an auxiliary equation of type $( * )$, where the right-hand side is a rank one matrix.