【作者】 乔云；

【导师】 吴化璋；

【作者基本信息】 安徽大学 ， 应用数学， 2007， 硕士

【摘要】 本文围绕Vandermonde矩阵和几种广义Vandermonde矩阵(如Cauchy-Vandermonde矩阵、合流Vandermonde矩阵、函数形式的合流Vandermonde矩阵以及q-adic Vandermonde矩阵)展开讨论，归纳总结了它们的若干性质。首先介绍了Vandermonde矩阵的概念和若干性质，如与多项式插值问题之间的联系，以及Vandermonde矩阵的三角分解等。其次，本文针对几种广义Vandermonde矩阵进行了归纳和推导。例如，Cauchy-Vandermonde矩阵在Lagrange类型插值问题中可以看作其系数矩阵。同时本文给出了Cauchy-Vandermonde矩阵的其它性质，如逆矩阵的表示等。此外，围绕合流Vandermonde矩阵，一方面给出它的插值意义，得到合流Vandermonde矩阵与Hermite插值问题之间的联系：另一方面给出行列式值的递推公式等。对于函数形式的合流Vandermonde矩阵和q-adic Vandermonde矩阵，同样得到了它们与相关插值问题的联系以及其它性质。最后，通过给定一个具有相同重度的结点序列，构造对应的合流Vandermonde矩阵和特殊的Hankel矩阵，给出了这两种矩阵之间的关系。此外，对应结点构造一种特殊的Toeplitz矩阵，类似可得到一系列相关性质。更多还原

【Abstract】 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.更多还原