三维常系数线性系统的标准基解矩阵公式
The Formula of the Elementary Solution Matrix of the 3-Dimensional Linear System
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摘要: 我们知道,对n维常系数线性系统X=AX,X(0)=η;其解可表达为。其中λj是A的nj重特征根,vj是(A-λj I)nj u=0的解空间中的元素,且η=v1+…+vk,由此可得到线性系统的基解矩阵expAt。直接使用这种方式计算不太方便,因为确定vj的过程很繁。本文给出了三维系统的基解矩阵的直接算法,expAt可直接由A的元素表出,从而为三维线性系统的分析计算带来方便。Abstract: It is well known that the solution of the n-dimensional constant coefficient linearsystem X=AX, X(0)=η can be represented as in which, for all j, λi, is the n-th characteristic root of A, and vj is the element ofthe solution space of (A-λj I)n ju=0 and η=v1 +… +vk. From this, we can getthe elementary solution matrix EXPAt of the system. However, this is not a conve-nient method if it is used directly in computation, because to find vj would involve acomplicated process. This paper gives a direct calculation method where by to solve theproblem of the 3-dimensional systems. The EXPAt can be represented by the elemen-ts of A, which will bring about convenient in calculation.
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