Research On Cauchy And The Related Structured Matrices

The structured matrix is an important topic in the area. of applied matrices. Cauchy(or generalized Hilbert)matrix is an important matrix among all structured matrices, and displacement structure theory and interpolation problems are related to Cauchy matrix, In this thesis, we mainly study these properties of Cauchy matrix and other related matrices, and describe the rational interpolation problems of Cauchy linear system. In addition, we also characterize a new type generalied matrix as paired Cauchy-Vandcrmonde matrix, and discuss the relative properties of Cauchy-Vandermonde matrix.The thesis is divided into three chapters.In Chapter 1, firstly, introduce the background of the structured matrices and the development of the theory of displacement structure. The ralations between structured matrix and some applications of structured matrix are dissaussed. Finally, formulate problems to investigate and resluts.In Chapter 2, we study the properties of Cauchy matrix, especially the property of displacement struture. Based on the discussion of displacement struture, we describe the properties of inversion formulas and the factorization forms, and give an fast algorithm for the paired Cauchy matrix based on LU factorization. On the basis of researchs of Heinig.et.al., we construct a new definition of paired Cauchy-Vandemonde matrix, and discuss the relative properties of paired Cauchy-Vandemonde matrix.In Chapter 3, we mainly study the interpolation problems of Cauchy matrix and other relative matrices, and give the interpolation interpretations. We will describe the interpolation formulas of the paired Cauchy-Vandemonde matrix from some different veiwpoints, and the relations between them.