Lyapunov, Bézout, and Hankel
Given a polynomial f of degree n, we denote by C its companion matrix, and by S the truncated shift operator of order n. We consider Lyapunov- type equations of the form \(X-SXC=W\) and \(X-CXS=W.\) We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C, for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.
scientific article; zbMATH DE number 3904764