當我們解一個遞迴關係的特徵方程式時，不管解得的根是相異根或者是重根，皆視這些跟產生的解之線性獨立為理所當然，因此很容易寫出此遞迴關係的通解乃是這些解的線性組合。在本文中，我們將透過廣義的范德蒙行列式(Generalized Vandermonde Determinant)的計算，很清楚地看出這些解之間的線性獨立。 / When we solve a characteristic equation of a recurrence relation, no matter what the roots are distinct or not, we take the linear independence of these solutions producing by each root for granted. Basing on this fact, we can easily write out the general solutions of this recurrence relation by using linear combination of these solutions. In this paper, we will see the linear independence of these solutions very clearly through the calculation of Generalized Vandermonde Determinant.

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.