| The study of circulant matrices, an important component of the matrix theory, has become one of the most important and active research fields in applied mathematics. Skew-symmetric and skew-circulant matrices studies are basilic part of circulant matrix. Due to the special features of skew symmetric and skew-circulant matrices, it's necessary for us to generalize their unique structures and characteristics, such as: all kinds of polynomial representations, diagonalization, special decomposition, nonsingularity eigenvalues, characteristic polynomial and fast algorithms for computing minimal polunomial, inverse, self-reflective g -inverse, group inverse and Moore-Penrose inverse, and so on. The main contents of this paper are as follows:Firstly, skew-symmetric and skew-circulant matrix was defined, its diagonalization by using Vandermonde matrix was discussed, through which some results were given. With these results, related features of skew-symmetric and skew-circulant matrix were deduced. And in this way, a simple algorithm of reversing of Skew-symmetric and skew-circulant matrices were given. Secondly, with the development of the skew symmetric and skew circulant matrix, several block skew-symmetric and skew-circulant matrix were defined, two of which were discussed in terms of features and characteristics. Finally, based on these characteristics , the eigenvalues and eigenvalues polynomials and its diagonal matrix were given.This article is divided into three parts:I: It gives the relevant background knowledge, mainly about the study of circulant matrices at home and abroad, the basic concepts, characteristics of circulant matrices, and the basic computing instruments which have been frequently used in matrix theory and matrix calculations.II: It gives a new matrix type-- the skew-symmetric and skew-circulant matrix and a series of the characteristics of this matrix, and comes up with its diagonalization using the Vandermonde matrix. Then it gives the reverse method of the skew-symmetric and skew-circulant matrix, and the characteristics of the inverse matrix.III: In this paper, the concepts of two different sub-block of block skew-symmetric and skew-circulant matrix was given, and their characters were discussed, relevant conclusions were presented. |
