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題名 | A Method for Solving the System Oflinear Equations by Applying Genera lized Inverses and the Properties of Their Solution Spaces=應用一般化反矩陣求線性方程組之通式解與其解空間特性之研究 |
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作者姓名(中文) | 林富森; | 書刊名 | 國立臺灣海洋大學人文數理學報 |
卷期 | 1 1991.10[民80.10] |
頁次 | 頁17-41 |
分類號 | 313.79 |
關鍵詞 | 反矩陣; 線性方程組; |
語文 | 英文(English) |
中文摘要 | 本文介紹一種應用一般化反矩陣解線性方程組之方法,和探討其解空間之特性。只要求出係數矩陣的一般化反矩陣即可求得方程組之通式解。此通式解為一特解與齊性解之和,所有解均可由此通式解產生。因此我們提出兩個演算法,一為求一般化反矩陣之演算法,另一為解線性方程組之演算法。我們發現解集合具有某些重要的特性,其一為不論是齊性或非齊性方程組,其解集合是一個凸集合,其二為非齊性方程組之解集合具有特殊之結構即它們的線性獨立解之線性組合並不一定是它的解,但只要其線性組合之係數和為一,則不論它是否線性獨立,任何數目之解的線性組合均是它的解。另外我們也提供一些判斷非齊性方程組是相容或矛盾之方法,且將它們加入解線性方程組之演算法中,使此演算法更具一般性。 |
英文摘要 | This paper presents a method for solving the system of linear equations by applying generalized inverse matrices and searches the properties of their solution spaces. This method can easily be used to solve the linear system only if a generalized inverse of the coefficient matrix can be found. We provide two algorithms, one for finding the generalized inverse, the other for solving the linear system. By this method, the solution set can be represented in a general form which is the sum of a particular solution and the homogeneous solution, and all the solutions can be generated by this form for a consistent system. We have found some important properties in the solution set. The solution set is convex whatever for homogeneous or non-homogeneous systems. The special structure of the solution set to the non-homogeneous system is that the linear combination of their linearly independent solutions may not be a solution except that the sum of their combination coefficients is unity, no matter how many dependent or independent solutions there are. We also propose some ways to examine the non-homogeneous system whether it is consistent or inconsistent, and put them into the algorithm for solving the linear system, so that the algorithm has much more generalization. |
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