Explicit Solutions of the Matrix Equation $\sum {A^i XD_i } = C$
Hȧrald K. Wimmer · SIAM Journal on Matrix Analysis and Applications · 1992
A module theoretic approach is developed to study the linear matrix equation $\sum {A^i } XD_i = C$. If the solution X is unique it can be expressed in the form $X = \sum {A^k CG_k } $. The matrices $G_k $ can be determined from an auxiliary equation which involves the companion matrix of the characteristic polynomial of A. A connection is made with realizations of the inverse of the associated polynomial matrix $D(x) = \sum {D_i } x^i $. The more general equation $\sum {A_i } XD_i \doteq C$ is discussed under the assumption that the matrices $A_i $ are pairwise commutative.