分块矩阵求逆及其应用

目录

摘要 (1)

引言 (2)

一、概述 (2)

二、分块矩阵的求逆及其应用 (5)

第一节２×２分块矩阵的可逆性存在条件和求逆公式及其应用

(5)

第二节 3×3分块矩阵的可逆性存在条件和求逆公式及其应用

(14)

结束语 (21)

分块矩阵求逆及其应用

李东生

（渤海大学数学系 辽宁 锦州 121000 中国）

摘要：对于分块矩阵，我们比较熟悉分块矩阵的乘法，而对于分块矩阵的求逆，经常遇到的是22⨯分块矩阵的逆的证明问题，很少涉及分块矩阵逆的计算，并且我们在实际问题中还会遇到33⨯分块矩阵（或更高阶的分块矩阵）的求逆问题，所以我们研究这样的分块矩阵的可逆性存在条件以及求逆公式显得很有意义。分块是否合理是分块矩阵运算是否简便的关键，所以本文开头便对分块方法做了总结。接着，本文研究了较为简单的22⨯分块矩阵的可逆性存在条件以及求逆公式，并予以证明，总结了研究方法，还深入探讨了22⨯分块矩阵中含有零块时的可逆性存在条件以及求逆公式。以22⨯分块矩阵的研究方法为基础，探讨研究了33⨯分块矩阵的可逆性存在条件以及求逆公式，并试证成功，还总结出研究更高阶分块矩阵求逆方法。此外本文不仅侧重理论研究，而且侧重于实际应用，在文中列举了大量典型的阶数较高的矩阵，对他们如何分块才能使求逆过程更为简单作出分析，并给出了求解过程，真正做到了“理论联系实际”。 关键字：分块方法，分块矩阵，逆矩阵，可逆条件

Begging the negative matrix to a matrix of the cent

and it ′s applying

Li Dongsheng

(Department of Mathsmatic Bohai University Liaoning Jinzhou 121000 China) Abstract: For a matrix of the cent, we relatively know with the multiplication of dividing a matrix. But for begging the negative matrix to a matrix of the cent, we usually meet is 2 the negative certificate problem of a matrix of cent of rank. It is seldom to involve to divide the calculation that a matrix inverse, and we also will meet in actual problem begging 3 the negative certificate problem of a matrix of cent of rank.(or a matrix of more high-level cent).So it is very meaningfully to study this character of inverse of existence condition of such a matrix of cent; to beg the negative formula whether cent is reasonable is the key of whether a matrix operation is simple. What is more, the beginning of thesis does the summary to a method of cent. Immediately, the thesis has studied simple 2 ranks to divide a piece of matrix and the existence condition of inverse character. Finally the thesis gives the evidence. The method has been given, and when

there are zero-pieces in a matrix, the character of inverse condition and begging the negative formula are explored in the 2 ranks to divide a piece of matrix. In the basis of research method of 2 rank to divide a piece of matrix, the character of inverse, and begging the negative formula in 3 ranks to divide a piece of matrix are successfully proved, and also be summed up the method of begging the negative .In addition of this, this thesis not only lays particular emphasis on the theories research, but also deals of high level matrix of typical model which are used in the thesis, and how they divide the piece to make begging negative process more simple is also be analyzed . The process of how to solve is also gi ven. “Theories contact actual” is real attained in this thesis.

key words: the method of dividing the matrix into pieces; a matrix of cent ; negative matrix ; the condition that the matrix has a negative matrix.

引言

我们在处理一些多元线性方程组时，常常用系数矩阵，而且一般情况下，它们的阶数较高，在求解过程中，我们还要常常要求它们的逆．若要用普通的初等变换法，或求伴随矩阵法求逆都很麻烦．这时我们就应该考虑用分块矩阵法求矩阵的逆．我们知道并不是所有的矩阵都有逆，我们要求逆就应该判断矩阵是否可逆，然后再求逆．本文首先介绍了分块矩阵的定义以及常用的分块方法，重点介绍２×２分块矩阵和３×３分块矩阵的可逆性存在条件，并给出了普遍使用的求逆公式，而且文中还举了一些有代表性的例题，并讨论是如何分块，如何应用求逆公式的．一概述

１．分块矩阵的定义

在处理级数较高的矩阵是常用矩阵分块的方法．我们可以把大矩阵看成是由小矩阵组成的，就如矩阵是由数组成的一样．特别在运算