政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/32565

政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/32565

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政大典藏 > College of Science > Department of Mathematical Sciences > Theses > Item 140.119/32565

Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/32565

Title:	有關廣義范氏矩陣的研究：其行列式、反矩陣、LU分解、及應用 Studies on Generalized Vandermonde Matrices: Their Determinants, Inverses, Explicit LU Factorizations, with Applications
Authors:	李宣助 Hsuan-Chu,Li
Contributors:	陳永秋李陽明 Eng-Tjioe,Tan Young-Ming, Chen 李宣助 Hsuan-Chu,Li
Keywords:	古典范氏矩陣廣義范氏矩陣全正廣義范氏矩陣行列式 LU分解 1-帶狀分解廣義范氏矩陣的反矩陣 Schur 函數 Kostka 數
Date:	2006
Issue Date:	2009-09-17 13:45:38 (UTC+8)
Abstract:	古典及廣義的范氏矩陣普遍存在於數學之中，而且最近有多位作者對於它們的行列式、反矩陣、LU分解及應用等做了各種的研究。在這篇論文中我們主要探討兩個主題：一是廣義范氏矩陣的回顧，二是廣義范氏矩陣的不同分解。在第一個主題，我們僅利用數學歸納法來證明兩種已知型態的廣義范氏矩陣行列式的公式，與之前錢福林及Flowe-Harris的證明方法截然不同。在構成本篇論文主要結果的第二個主題中，我們致力於兩個目標：首先，我們探討某一特殊類的廣義范氏矩陣之轉置矩陣且成功地得到它的LU分解並將其明確地表示出來。更進一步地，我們將LU分解表示成1-帶狀矩陣的乘積並得到它的反矩陣。其二，我們考慮全正廣義范氏矩陣且在不使用Schur函數的情況下得到它唯一的LU分解，此結果優於Demmel-Koev需用到Schur函數的結果。同時，我們也得到該矩陣的行列式及反矩陣並將Schur函數明確地表示出來。基於上述結果，藉著將Schur函數展開，我們獲得一種計算Kostka數的方法。 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.
Reference:	[1] A. Björck, T. Elfving, Algorithms for confluent Vandermonde systems, Numer. Math. 21 (1973) 130-137. [2] A. Björck, V. Pereyra, Solution of Vandermonde systems of equations, Math. Comp. 24 (1970) 893-903. [3] Young-Ming Chen, Hsuan-Chu Li and Eng-Tjioe Tan, An explicit factorization of totally positive generalized Vandermonde matrices avoiding Schur functions, to appear in Applied Mathematics E-Notes. [4] J. Demmel, P. Koev, The accurate and efficient solution of a totally positive generalized Vandermonde linear system, SIAM J. Matrix Anal. Appl., Vol. 27, No. 1 (2005) 142-152. [5] M. El-Mikkawy, A. Karawia, Inversion of general tridiagonal matrices, Applied Mathematics Letters 19 (2006) 712-720. [6] Randolph P. Flowe, Gary A. Harris, A note on generalized Vandermonde determinant, SIAM J. Matrix Anal. Appl. Vol. 14, No. 4 (1993) pp. 1146-1151. [7] R. P. Grimaldi, Discrete and Combinatorial Mathematics, 2nd Ed., Addison-Wesley, Reading, MA, 1989. [8] Jian-min Han, The calculation of the generalized Vandermonde determinant, Journal of Baoji University of Arts and Sciences(Natural Science). Vol. 25. No. 1. (2005) 26-28 (in Chinese). [9] S. Lang, Linear Algebra, 3rd ed., Springer-Verlag, New York, 1987. [10] H. C. Li, A proof about the generalized Vandermonde determinant, Master Thesis, National Chengchi University, Mucha, Taipei, Taiwan (1994). [11] Hsuan-Chu Li, Eng-Tjioe Tan, On a special generalized Vandermonde matrix and its LU factorization, to appear in Taiwanese Journal of Mathematics. [12] C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill Book Company (1968). [13] Ronald E. Mickens, Difference Equations, Van Nostrand Reinhold Company, Inc., New York, 1987. [14] Janvier Nzeutchap, Frédéric Toumazet and Franck Butelle, Kostka numbers and Littlewood-Richardson coefficients: Distributed computation, Preprint (2007). [15] H. Oruç, G. M. Phillips, Explicit factorization of the Vandermonde matrix, Linear Algebra and Its Applications 315 (2000) 113-123. [16] Fulin Qian, Generalized Vandermonde determinant, Journal of Sichuan University Natural Science Edition Vol. 28, N0.1 (1991) pp. 36-40. [17] W. P. Tang, G. H. Golub, The block decomposition of a Vandermonde matrix and its applications, BIT 21 (1981) 505-517. [18] Richard P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1990. [19] Loring W. Tu, A generalized Vandermonde determinant, arXiv:math.AC/0312446 v1 24 Dec 2003. [20] Luis Verde-Star, Inverses of generalized Vandermonde matrices, Journal of Mathematical Analysis and Applications 131, (1988) pp. 341-353. [21] Sheng-liang Yang, On the LU factorization of the Vandermonde matrix, Discrete Applied Mathematics 146 (2005) 102-105. [22] Sheng-liang Yang, Generalized Vandermonde matrices and their LU factorization, Journal of Lanzhou University of Technology, 2004 Vol. 30 No. 1 P. 116-119 (in Chinese, 廣義Vandermonde矩陣及其LU分解). [23] Y. Yang, H. Holtti, The factorization of block matrices with generalized geometric progression rows, Linear Algebra and Its Applications 387 (2004) 51-67.
Description:	博士國立政治大學應用數學研究所 90751502 95
Source URI:	http://thesis.lib.nccu.edu.tw/record/#G0090751502
Data Type:	thesis
Appears in Collections:	[Department of Mathematical Sciences] Theses

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