| Summary: | 博士 === 國立政治大學 === 應用數學研究所 === 95 === Classical and generalized Vandermonde matrices are ubiquitous in mathematics, and various studies on their
determinants, inverses, explicit LU factorizations with
applications are done recently by many authors. In this thesis we shall focus on two topics: One is generalized Vandermonde matrices revisited and the other is various decompositions of some generalized Vandermonde matrices. In the first topic, we prove the well-known determinant formulas of two types of generalized Vandermonde matrices using only mathematical induction, different from the proofs of Fulin Qian's and Flowe-Harris'. In the second
topic, which constitutes the main results of this thesis, we
devote ourself to two themes. Firstly, we study a special class which is the transpose of the generalized Vandermonde matrix of the first type and succeed in obtaining its LU factorization in an explicit form. Furthermore, we express the LU factorization into 1-banded factorizations and get the inverse explicitly. Secondly, we consider a totally positive(TP) generalized Vandermonde matrix
and obtain its unique LU factorization without using Schur
functions. The result is better than Demmel and Koev's which is involved Schur functions. As by-products, we gain the determinant and the inverse of the required matrix and express any Schur function in an explicit form. Basing on the above result, we obtain a way to calculate Kostka numbers by expanding Schur functions.
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