| Bezoutian matrix is important to determine the stability of linear systems as a tool.In recent years, the study of Bezoutian matrix is deeply with the development of systems and control theory.Classical Bezoutian matrix is also promoted in a number of aspects . Polynomial Bezoutian matrix is an important research direction. In [3] Z.H.Yang gave a general notion of polynomial Bezoutian matrix, and presented: (1)generalized Barnett decomposition formula of polynomial Bezoutian matrix;(2) an intertwining relation of polynomial Bezoutian matrix and Alliance matrix ; (3) the generalized Bezoutian reduction via confluence Vander monde matrix.These conclusions are similar to classical Bezoutian matrix.In [1] G.Heinig and K.Rost also studied many other properties of the classical Bezoutians.In this paper, I firstly give some important conclusions of polynomial Bezoutian matrix.The first part of the job is to study properties of polynomial Bezoutian matrix with purely algebraic methods.With easier ways of presenting the proof of three known conclusions of polynomial Bezoutian matrix, but also give some new properties of the polynomial Bezoutian matrix, and discuss the solutions of Lyapunov type equation, too.In the second part, we firstly introduce the first concept of the Toeplitz-Bezout matrix, and secondly we study some properties of Toeplitz-Bezout matrix with the method of polynomial model and the matrix representation of operator.We also prove the fact that the Toeplitz-Bezout matrix is the matrix representation of a pair of duality basis.At the same time, we give:(1)the factorization formula of the Toeplitz-Bezout matrix; (2) an intertwining relation of Toeplitz-Bezout matrix and the class of the companion matrix; (3)introduce the resultant matrix about a pair of polynomials and give the relation between the resultant matrix and Toeplitz-Bezout matrix.Then we take advantage of this relation, and get triangular decomposition formula of Toeplitz-Bezout matrix. |
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