Solutions of the matrix linear bilateral polynomial equation and their structure

Abstract

We investigate the row and column structure of solutions of the matrix polynomial equation \[ A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda), \] where \(A(\lambda)\), \(B(\lambda)\) and \(C(\lambda)\) are the matrices over the ring of polynomials \(\mathcal{F}[\lambda]\) with coefficients in field \(\mathcal{F}\). We establish the bounds for degrees of the rows and columns which depend on degrees of the corresponding invariant factors of matrices \(A (\lambda)\) and \( B(\lambda)\). A~criterion for uniqueness of such solutions is pointed out. A method for construction of such solutions is suggested. We also established the existence of solutions of this matrix polynomial equation whose degrees are less than degrees of the Smith normal forms of matrices \(A(\lambda)\) and \( B(\lambda)\).