| dc.contributor.advisor | 陳永秋<br>李陽明 | zh_TW |
| dc.contributor.advisor | Eng-Tjioe,Tan<br>Young-Ming, Chen | en_US |
| dc.contributor.author (Authors) | 李宣助 | zh_TW |
| dc.contributor.author (Authors) | Hsuan-Chu,Li | en_US |
| dc.creator (作者) | 李宣助 | zh_TW |
| dc.creator (作者) | Hsuan-Chu,Li | en_US |
| dc.date (日期) | 2006 | en_US |
| dc.date.accessioned | 17-Sep-2009 13:45:38 (UTC+8) | - |
| dc.date.available | 17-Sep-2009 13:45:38 (UTC+8) | - |
| dc.date.issued (上傳時間) | 17-Sep-2009 13:45:38 (UTC+8) | - |
| dc.identifier (Other Identifiers) | G0090751502 | en_US |
| dc.identifier.uri (URI) | https://nccur.lib.nccu.edu.tw/handle/140.119/32565 | - |
| dc.description (描述) | 博士 | zh_TW |
| dc.description (描述) | 國立政治大學 | zh_TW |
| dc.description (描述) | 應用數學研究所 | zh_TW |
| dc.description (描述) | 90751502 | zh_TW |
| dc.description (描述) | 95 | zh_TW |
| dc.description.abstract (摘要) | 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。 | zh_TW |
| dc.description.abstract (摘要) | Classical and generalized Vandermonde matrices are ubiquitous in mathematics, and various studies on theirdeterminants, inverses, explicit LU factorizations withapplications 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 secondtopic, which constitutes the main results of this thesis, wedevote 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 matrixand obtain its unique LU factorization without using Schurfunctions. 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. | en_US |
| dc.description.tableofcontents | Abstract i中文摘要 iiIntroduction 1Part I Generalized Vandermonde Matrices Revisited 4Chapter 1 Ubiquity of Vandermonde Matrices 5Chapter 2 An Inductive Proof on the Determinant of the Generalized Vandermonde Matrix 15 2.1 The Casoratian of Functions •••••••••••••••••••••••••15 2.2 The Determinant •••••••••••••••••••••••••••••••••••••172.3 The Example •••••••••••••••••••••••••••••••••••••••••24Chapter 3 On Flowe-Harris’Proof of the Determinant Formulaof the Generalized Vandermonde Matrix 303.1 The Determinant•••••••••••••••••••••••••••••••••••••••••••••••303.2 The Example••••••••••••••••••••••••••••••••••••••••••37Part II Various Decompositions of Some Generalized Vandermonde Matrices 43Chapter 4 On a Special Generalized Vandermonde Matrix and Its LU Factorization 444.1 The LU Factorization of $V_{\\{n;1,n-1\\}}$••••••••••••454.2 Factorization of $V_{\\{n;1,n-1\\}}$ into 1-Banded(Bidiagonal) Matrices•••••••••••••••••••••••••••••••••••••534.3 Applications to the Closed-form Formula of $V_{\\{n;1,n-1\\}}^{-1}$••••••••••••••••••••••••••••••••••••••••••••••••64Chapter 5 An Inductive Computation of Totally Positive Generalized Vandermonde Matrices Avoiding Schur Functions 755.1 The LU Factorization of $G_n$••••••••••••••••••••••••775.2 The Determinant of $G_n$•••••••••••••••••••••••••••••835.3 The Inverse of $G_n$•••••••••••••••••••••••••••••••••87 5.4 The Calculation of Schur Function••••••••••••••••••••99 5.5 An Application to Kostka Numbers••••••••••••••••••••104Bibliography 111Appendix 114 | zh_TW |
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| dc.language.iso | en_US | - |
| dc.source.uri (資料來源) | http://thesis.lib.nccu.edu.tw/record/#G0090751502 | en_US |
| dc.subject (關鍵詞) | 古典范氏矩陣 | zh_TW |
| dc.subject (關鍵詞) | 廣義范氏矩陣 | zh_TW |
| dc.subject (關鍵詞) | 全正廣義范氏矩陣 | zh_TW |
| dc.subject (關鍵詞) | 行列式 | zh_TW |
| dc.subject (關鍵詞) | LU分解 | zh_TW |
| dc.subject (關鍵詞) | 1-帶狀分解 | zh_TW |
| dc.subject (關鍵詞) | 廣義范氏矩陣的反矩陣 | zh_TW |
| dc.subject (關鍵詞) | Schur 函數 | zh_TW |
| dc.subject (關鍵詞) | Kostka 數 | zh_TW |
| dc.title (題名) | 有關廣義范氏矩陣的研究：其行列式、反矩陣、LU分解、及應用 | zh_TW |
| dc.title (題名) | Studies on Generalized Vandermonde Matrices: Their Determinants, Inverses, Explicit LU Factorizations, with Applications | en_US |
| dc.type (資料類型) | thesis | en |
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