Research On Toeplitz Bezoutian And Bezoutian With Respect To A General Basis

With the wide applications in the inverse of structured matrices and the stability of linear control systems, Bezout matrix has become an important research subject in the theory of matrices and operators. The study of classical Bezout matrix has gradually fulfilled. This paper mainly extends Bezout matrices to a more general case and also studys the T-Bezoutians and Bezout matrices under a general basis.In this paper, the backgrounds of Bezout matrices are given in part one. In part two, some properties of T-Bezout matrices under a general basis are derived via the properties of T-Bezout matrices under the power basis by using operator approach method. Such properties include T-Bezoutians under a general basis being the matrix representation of w-operator relative to a pair of dual bases; an intertwining relation between T-Bezout matrix and confederate matrix under a general basis; the generalized Barnett factorization formula; another representation of Vandermonde matrix and its relation with controllability and observability matrices. In part three, the properties of Bezout matrices under a general basis is mainly introduced using the connection between Bezout matrices under the power basis and controllability/observ-ability matrices. Part four gives the Bezout matrices under the Lagrange basis and their relationship with the Bezout matrices under the power basis.