The Problem Of Toeplitz Matrices Over A Ring And A Complex Field

Matrix is indispensable as a modern mathematical tool in natural science, engineering and even in many areas of social science. Toeplitz matrix is a matrix who has attracted much attention because of its widely useness in many fields. Toeplitz matrix was first put forward at the beginning of twentith century by the German mathematician Tuopulieci (Toeplitz Otto).Toeplitz matrix and the calculation of the correlation matrix has been a hot research topic at home and abroad. Particular Toeplitz matrix has a wide range of applications in biology, physics,mathematics.Therefore, the study of Toeplitz matrix was necessary.From the current view of many domestic and foreign literatures,research on the Toeplitz matrix and its application is compelling not only in the present, but in the future.Toeplitz matrices are characterized by that the main diagonal elements are equal to each other and elements parallel to the main diagonal are equal to each other. Generally the inverse of a Toeplitz matrix is no longer a Toeplitz matrix,To give the inverse of a Toeplitz matrix is with certain difficulties.In the first part of the paper by making use of the known results for inverse matrix of the Toeplitz matrix,We make a further research for the inverse matrix of Toeplitz matrix over a class of ring with unit element. We give the reversible conditions of Tn when given Tn-1 is reversible.If Tn-1 is reversible,and (?) or (?) is reversible, then Tn is reversible, and (?) and (?) are all reversible.In The latter part of the paper we study the nature of anti-cyclic matrix and unit Toeplitz matrix. We give the sufficient conditions on which elements made up of a cyclic algebra and on which elements made up of anti-cyclic algebra.That is,we describes a class of recycling Toeplitz and anti-cyclic Toeplitz linear space over complex field, which are linear space respectively formed by anti-symmetric matrix exchanged with unit circle matrix and unit anti-cyclic matrix and the anti-hermite matrix exchanged with unit circle matrix and unit anti-cyclic matrix.Furthermore we characterize the base of cyclic Toeplitz linear space over complex field. All of the above we did is to explore the structure and nature of the Toeplitz matrix for further promotion.