Research On Properties Of Bezout Matrix And Toeplitz-Bezout Matrix For Polynomial Matrices

An important item of systems analysis is the control system stability analysis, Bezout matrix is an powefull tool to research of the stability of linear systems. With the development of control theory in recent years, the concept of Bezout and its various generalizations play an important role in modern linear algebra, which attraches scholars'enough attention, and draws lots of new results. Therefore, the study of the characteristics of Bezout matrix has important practical significance.In this dissertation, we discuss Hankel-Bezout (for short Bezout) matrix for polynomial matrices, and give its definition and properties. We can show the Bezout matrix is a matrix representation of a polynomial model homomorphism with respect to a particular dual pair of bases, and intertwining relation with other matrices similar to that of the scalar is presented. In tensor space of the polynomial model, the conditions for the solution of the homogeneous polynomials Sylvester equation is exactly this type of matrix; under this basis, we give the definition of the Toeplitz-Bezout (for short T-Bezout) matrix for the polynomial matrices, and which it is a matrix representation of a polynomial model homomorphism with respect to a dual pair of bases. Finally, the solution of this class homogeneous polynomial Stein equation is such matrices. This paper includes three chapters as below:The first chapter describes the backgrounds and significance of the problems. And the main work done in this article are introduced.The second chapter firstly gives the basic properties of Bezout for polynomial matrices and presents the reprentation of this Bezout by the method of polynomial model, in addition the intertwining relation with each other matrices are deduced. Finally, we discuss the solutions of the homogeneous polynomials Sylvester equation.In the last chapter we deal with the case of T-Bezout for polynomial matrices.