有理矩阵有理相似对角化的计算机实现  

Using the Computer to Realize Rational Matrix Similarity Diagonalization

作　　者：王纯[1] 周腾锦[1] 

机构地区：[1]韩山师范学院数学与应用数学系

出　　处：《价值工程》2013年第16期194-199,共6页Value Engineering

基　　金：2012年国家大学生创新创业训练项目论文;项目名称:有理矩阵有理对角化问题的算法及程序设计研究;项目编号:1057812034

摘　　要：人工计算有理矩阵能否在有理数域上相似对角化是非常困难的,所以需要计算机来辅助实现,然而已有的数学软件对此问题的计算结果却存在着误差,于是需要研究有理矩阵在有理数域上相似对角化的算法及程序.因此在直接进行分数运算的基础上,首先使用矩阵的幂与来计算有理矩阵的特征多项式,克服了直接计算行列式的方法所存在的算法设计上的困难,其次根据有理多项式有理根的求法计算出有理矩阵的有理特征根,进而精确地计算出相应的有理特征向量,从而成功设计出判断及实现有理矩阵在有理数域上对角化的算法及相应的语言程序,使用该程序能够精确地解决有理矩阵在有理数域上相似对角化的问题。Manual computation of wether the rational matrix can be similar diagonalization in the rational number field is very difficult, so we need computer to assist in the implementation.However, the pre-existing mathematical software calculation results of the problem have some errors, so we need to study the algorithm and program of rational matrix in rational number field similarity diagonalization. So on the basis of directly proceeding the fractional arithmetic. Firstly we use the matrix power and trace to calculate the characteristic polynomial of rational matrix, and overcome the algorithm design difficulty of directly computing the determinant method, secondly according to the rational root of rational polynomial method,calculate the rational characteristic root of a rational matrix, and then precisely calculate the corresponding rational characteristic vector, Thereby we successfully designed the algorithm and the corresponding language program for rational matrix diagonalization in rational number field.Using the program is able to accurately solve the rational matrix diagonalization similar problems in the rational number field.

关 键 词：有理矩阵 有理相似对角化 算法 程序 

分 类 号：O151.21[理学—数学]