Using Matrices to Solve a System of Simultaneous Homogeneous Differential Equations with Constant Coefficients

使用矩陣以求解聯立常數係數齊性微分方程組

廖秋雄(Chiu-Hsiung Liao)

《勤益學報》 25期 (2007/12) Pp. H-1-H-15

https://doi.org/10.6753/CYJ.200712.0013

摘要

在自然科學之有趣的領域裡，包括數學、物理及工程學等，充滿著許多數學問題。不論幾何意義或是物理定律都經由數學模式來解讀和認知。一但自然界的問題呈現出來，最佳的討論的方式就是將之模式化為數學問題。當數學問題得到解答，所面對的問題也得以解決。
大多數我們所建構的數學模式都與微分方程式密不可分。因此，處理自然界的問題與解微分方程式是息息相關的。若出現聯立微分方程組時，我們不利用矩陣而嘗試直接來求解方程組的話，那是非常累人的。相反地，如果使用矩陣及其相關運算來解微分方程組就顯得簡單、有效率和值得讚許。
在本篇論文，我們展示如何應用矩陣的精髓及基本運算來解微分方程組。其結果揭露了這些方法是可信賴的，同時也展示矩陣在線性代數這方面的效力。

關鍵字

固有值 ； 核 ； 可對角化矩陣及Jordan典型矩陣

並列摘要

In the interesting areas of natural science, including mathematics, physics, and engineering etc., there are full of mathematical problems. Not only are geometric meaning, but also physical rules are explained and recognized through mathematics models. Once a problem in natural world comes up, the best way to go through is modeling it into a mathematics model. When the mathematics problem is solved, the confronting issue is dealt.
Most of the mathematics models that we set up are concerned the differential equations. So, to deal with our problems in natural world, it is crucial to solve differential equations. If a system of differential equations is met, it is labor to solve it straightly without using matrices operations. On the contrary, it is easily, efficiently and praiseworthily to find a solution to a system of differential equations through matrices and the operations on them. In this paper, we display how to use the essence of matrices and the fundamental operations to solve a system of differential equations. The bottom line not only reveals the methods being reliable, also demonstrates this part in linear algebra being powerful.

並列關鍵字

eigenvalue ； kernel ； diagonalizable matrix ； and Jordan canonical form matrix