Some Problems About Generalized Vandermonde Matrices

In this thesis, we introduce the Vandermonde matrix and several generalizedVandermonde matrices(Cauchy-Vandermonde matrix, confluent Vandermonde matrix,functional confluent Vandermonde matrix and q-adic Vandermonde matrix), anddiscuss some important properties of these matrices.First, we introduce the Vandermonde matrix, and give its relationship topolynomial interpolation problem. Then we discuss its other properties such astriangular decomposition.Secondly, we sum up some generalized Vandermonde matrices, and describetheir properties. For example, we interpret Cauchy-Vandermonde matrix as thecoefficient matrix of a Lagrange interpolation problem, and give representation ofinversion. We discuss the confluent Vandermonde matrix, and give its relationship toHermite interpolation problem. Then, we give some similar properties of thefunctional confluent Vandermonde matrix and q-adic Vandermonde matrix.Finally, we give the finite sequences of interpolation nodes in which each nodehas the same multiplicity, construct its corresponding confluent Vandermonde matrixand Hankel matrix, and prove the relationship between them. Also, we prove thesimilar properties of Toeplitz matrix.