Study On The Theory Of Vandermonde Matrices And Its Applications

Vandermonde matrix has been widely used in the fields of mathematics and engineering. As a kind of important matrix in scientific computation, it has aroused the attention of many mathematics workers. We recall firstly the properties of classical vandermonde matrix and its applications under the standard power basis and then discuss successively the inversion of Vandermonde matrix. The connections with the theory of interpolation are also discussed, meanwhile the inverse matrix can be obtained by the theory of interpolation.The first part of this article reviews the development history and application background of the Vandermonde matrix.The second chapter is used to discuss the Bezout matrix which is closely related to the Vandermonde matrix. The relationships between these two classes of matrices are then studied by a new method, called the operator shift. The applications of Vandermonde matrix in companion matrix and Hankel matrix are given. Successively, the third part is devoted to discuss the classical Vandermonde matrix under the satandard basis and their applications in interpolation theory. The Homer and Lagrange polynomials are used to solve the inversion of Vandermonde matrix in linear system.As an important application, we do some investigations on the connections with Pascal matrix and Stirling matrix. In the last part we mainly studies the relationship with stochastic matrix in terms of the Pascal matrix and Stirling matrix. At the same time, the definition of Stirling matrix and Pascal matrix under q-integral is given. Explorations on the relationship among these special matrix and combinations are arranged in the end of the paper.