Research On The Properties Of Toeplitz-Bezout Matrices And The Inversions Of Infinite Generalized Block Toeplitz Matrices

Research on the properties of Bezout matrices and Toeplitz matrices is an important topic in the field of the matrix and operator theory, which is closely connected with modern equation theory, polynomial stability theory, system control theory and the interpolation problems. Therefore, it has academic and practical meanings to study their properties.This thesis presents systematic research on the properties of T-Bezout matrices with respect to a standard basis and a general polynomial basis respectively. New methods for the inversions of infinite generalized block Toeplitz matrices are also discussed.First, we introduce the backgrounds of the T-Bezout matrices and Toeplitz matrices and the main work done in this thesis.Next, many significant properties of T-Bezout matrices are deduced. We use the pure algebraic methods to discuss its properties such as triangular decomposition, Barnett factorization formula, intertwining relations, T-Bezoutian reduction via Vandermonde matrix and the relations with resultant matrix, etc. Subsequently, through the relation between a standard basis and a general basis, the similar properties of polynomial T-Bezoutian matrices are obtained.Finally, the algorithms for inversions of the infinite generalized Toeplitz matrices are investigated. Based on the theory of displacement structure we expand the Ben-Artzi-Shalom's approach for finite Toeplitz matrices to inverse for the infinite generalized block Toeplitz matrices, and obtain a new method for the inversion of infinite generalized block Toeplitz matrices.