Generalized Bezoutian and matrix equations

Abstract

A natural generalization of the classical Bezout matrix of two polynomials is introduced for a family of several matrix polynomials. The main aim of the paper is to show that this generalized Bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials M(λ)Y(λ) + Z(λ)L(λ) = R(λ) and the class of linear matrix equations AX - XB = C. Each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized Bezoutian corresponding to a solution (Y(λ), Z(λ)) of the equation in matrix polynomials is a solution of the matrix equation, and conversely, any solution X of the matrix equation is a generalized Bezoutian corresponding to a certain solution of the equation in matrix polynomials. In particular, either both equations are solvable or both have no solutions. Explicit formulas connecting the solutions of the two equations are given. Also, various representation formulas for the generalized Bezoutian are derived, and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed. © 1988.