Generalized Bézoutian and matrix equations (Q1097329)

Let \(L_ r(\lambda)\) and \(M_ r(\lambda)\) be \(n\times n\) matrix polynomials satisfying \(M_ r(\lambda)L_ r(\lambda)=0.\) Using \((x- y)^{-1}M_ r(x)L_ r(y)=\Sigma x\quad iy\quad j\Gamma_{ij},\) the generalized Bézoutian \(B=(\Gamma_{ij})\) is defined and it is shown that this generalized Bézoutian serves as a connecting link between the class of equations in matrix polynomials \(M(\lambda)Y(\lambda)+Z(\lambda)L(\lambda)=R(\lambda)\) and the class of linear matrix equations \(AX-XB=C\). That is, given an equation in matrix polynomials \(M(\lambda)Y(\lambda)+Z(\lambda)L(\lambda)=R(\lambda)\) where L(\(\lambda)\) and M(\(\lambda)\) are monic matrix polynomials and \(\deg R\leq \deg \quad L+\deg M,\) then for any solution pair (Y(\(\lambda)\), Z(\(\lambda)\)) with \(\deg \quad Y\leq \deg Y\leq \deg L,\deg Z\leq \deg M,\) one can associate the generalized Bézoutian B such that \(\hat C_ MB- BC_ L=R^{(0)}.\) Conversely, any solution X of the matrix equation \(\hat C_ MX-XC_ L=R^{(0)}\) is the generalized Bézoutian for certain matrix polynomials. In particular, either both equations are solvable or both have no solutions.