Research On Polynomial Bezout Matrices

Bezout matrix has become a more and more important research subject in the theory of matrices and operators with its wide applications in many practical areas. In this thesis, we study the properties and operator representations of so called r-Bezout matrix generated by more than two polynomials.In Chapter1the backgrounds of Bezout matrices is introduced. In Chapter2summerizes some important properties of classical Bezout matrices and some generalizations. In Chapter3, we first give the definition of r-Bezout matrix and discuss some properties under the standard power basis, after that we extend it to the general polynomial basis. The generalized resultant matrix from the polynomial column vector and its connection with the r-Bezout matrix are investigated. Two different cares are considered. One is, there exists a greatest common divisor and the other is that, there is not any greatest common divisor but they can be factorized into some divisors multiplication. In the last parts, we establish some relationship between r-Bezout matrix and generalized companion matrix and discuss three cases of matrix in terms of the different representations of a(λ) and b(λ). In Chapter4, we first discuss the operator theory of generalized Toeplitz matrix, which can be seen the operator representation of a function of shift operator under a pair of dual bases. In the second part, we deduce some well known inversion formulas for the Toeplitz matrix by a unified polynomial approach. We conclude that the different inversion formulas can be regarded as seen the special cases in the frame of unified polynomial model.