Rational and polynomial solutions to 2-D polynomial matrix equations (Q1197573)

This paper studies the solution problem to the 2-\(D\) polynomial matrix equation \(AX+BY=C\), which occurs quite frequently in the study of discrete linear control systems. The solution is found by rewriting the matrix equation and applying the concept of left divisor. Explicit formulas for a rational solution \(X\) and \(Y\) are given. Necessary and sufficient conditions for the existence of a polynomial solution to the equation are also presented. A numerical example is given to illustrate the main results. It is also noted that similar results can be extended to \(n-D(n>2)\) polynomial matrix equations.