矩阵的分解及应用摘要一般而言，矩阵分解包括加法分解（将一矩阵分解为若干矩阵之和）及乘法分解（将一矩阵分解为若干矩阵之积）.矩阵分解的问题是伴随着求解线性方程组而提出的，许多矩阵的分解被用来求解线性方程组，来达到降低求解线性方程组解的计算复杂度的目的.此外，将矩阵分解为形式比较简单或性质比较熟悉的一些矩阵的和或乘积，这些分解式能够明显反映出原矩阵的有关数值特性，如矩阵的行列式、秩、特征值等.本文主要从矩阵分解的角度，对矩阵的初步理论进行较系统的归纳总结.首先介绍一般矩阵的加法分解和乘法分解，其次以可逆阵、对称阵、正定阵和对合阵为例讨论特殊矩阵的分解.关键词:矩阵；等价分解；相似分解；合同分解；QR 分解1Decomposition and Application of MatrixABSTRACTGenerally speaking，matrix decomposition includes addition decomposition(a matrix is decomposed into the sum of several matrices)and multiplicative decomposition(a matrix is decomposed into the product of several matrices).The problem of matrix decomposition is presented with the solution of linear equations ， the decomposition of many matrices is used to solve linear equations，to reduce the computational complexity of solving linear equations；in addition，the decomposition of a matrix into the sum or products of matrices that are simpler in form or more familiar in nature ， these factorization equations can clearly reflect the related numerical characteristics of the original matrix ， like the determinant of a matrix 、 rank 、 proper value of a matrix and so on. This paper mainly from the angle of matric decomposition，the preliminary theory of matric is summarized systematically. Firstly ， the addition and multiplication decomposition of general matrices are introduced，secondly，the decomposition of special matrices is discussed with the examples of invertible matrix，symmetric matrix，position definite matric and involute matric.Keyword ： matrix ； equi...
