常系数齐次线性常微分方程的解

Solution of System of Homogeneous Linear Ordinary Differential Equations With Constant Coeffient

本文首先证明了若当标准形矩阵有n个线性无关的循环向量 ,接着证明了常系数齐次线性常微分方程组存在m个与它的系数矩阵的m重特征根对应的线性无关的解。最后证明了常系数齐次线性常微分方程组存在n个线性无关的解 。

This paper first proves that Jordan's canonical matrix of order n has n linearly independant cyclic vector, and then proves that system of homogeneous linear ordinary differential equations with constant coeffients has m linearly independant solutions which correspond to m-ple eginvalues of the coeffient matrix. Finally it proves that system of homogeneous linear ordinary differetial equations with constant coeffients has n linearly independant solutions and arbitrary solution is linear combination of the n slutions .