逆矩阵的一些常见求法

Some Common Methods for Calculating the Inverse of Matrix

矩阵求逆的理论方法不仅在数学自身,而且在自动化、系统控制等领域有着广泛的实际应用。文章在逆矩阵、哈密尔顿—凯莱定理、线性方程组等有关知识的基础上,讨论了多种不同条件下矩阵逆的存在性及其求解公式。总结了求矩阵的逆的一些常见方法:定义法、伴随矩阵法、初等行变换法、初等列变换法、行列初等变换的结合法、哈密尔顿一凯莱定理法、三角矩阵法、线性方程组法,行列式法和分解矩阵法,并对每种方法,都给出了算例。

The calculating of the inverses of matrices is applied widely in many fields besides mathematics itself,such as the automation,control system and so on.In this thesis,on the basis of the knowledge of inverse matrix,Hamilton-Cayley theorem,linear equations and other relevant theory,we study the existence of inverse of matrices and give explicit formulas of in verse matrices under different conditions.Some common methods for calculating the inverse of matrix are summarized,including methods of making use of the definition;making use of the adjoint matrix;making use of elementary row transformation,elementary column transforma tion,the combination of elementary row transformation and elementary column transformation; making use of Hamilton-Cayley theorem,trianguar matrix,linear equations,determinant and the decomposition of matrix.We respectively discuss invertible conditions and give the formulas of the inverse of matrices when they are invertible.Numerical examples for corresponding methods are also given.