分块矩阵广义逆的研究

发布时间：2018-04-02 09:55

本文选题：分块矩阵广义逆　切入点：Banachiewicz-Schur形式　出处：《北京交通大学》2017年硕士论文

【摘要】：矩阵的广义逆理论一直都是世界矩阵论领域中一个非常重要的讨论分支,并且在工程运算求解线性方程组的一般解、最小二乘解以及最优化控制等研究中,广义逆理论都起着不可忽视的作用。对矩阵广义逆的研究,我们通常采用将矩阵分块成2 × 2的分块矩阵思想,通过研究其四个子块得到原矩阵的广义逆的相关性质。经过学者的广泛研究,得到的分块矩阵广义逆的表达式形式多样,但当运用到实际计算一般数阵的MP-逆问题上仍然具有很大的困难。本文将通过矩阵广义逆的可加性及两矩阵差的秩为零则这两个矩阵相等的性质给出了2 × 2分块矩阵广义逆的新表示方法。首先,通过推广广义逆的相关性质分别得到带有三个和带有两个零子块的2 × 2分块矩阵MP-逆表达式,在此基础上采用矩阵广义逆的可加性得到带有一个零子块及不带有零子块的分块矩阵MP-逆表达式。其次,研究了矩阵的Banachiewicz-Schur广义逆形式与矩阵{1}-逆、{1,2}-逆、{1,3}-逆、{1,2,3}-逆、{1,4}-逆,{12,4}-逆之间的等价条件,为矩阵的各种广义逆的表达式提供了一种新的思路。并将所得结论推广到了Hermit空间,得到了分块Hermit矩阵的Banachiewicz-Schur广义逆形式与其各种广义逆之间的等价条件。木章最后还研究了分块矩阵的Banachiewicz-Schur加权广义逆形式与其{1,3X}-加权逆、{1,2,3X}-加权逆、{1,4Y}-加权逆、{12,4Y}-加权逆之间的等价条件。最后,采用矩阵分解思想,研究了加边矩阵的MP-逆的表示方法,由此可以得到一种新的求解一般数阵广义逆的方法。
[Abstract]:The generalized inverse theory of matrices has always been a very important branch of discussion in the field of matrix theory in the world. The generalized inverse theory plays an important role. In the study of generalized inverse of matrices, we usually use the idea of dividing matrices into 2 脳 2 blocks. The related properties of the generalized inverse of the original matrix are obtained by studying its four subblocks. After extensive research by scholars, the expressions of the generalized inverse of the partitioned matrix are obtained in various forms. However, it is still very difficult to calculate the MP-inverse problem of the general number matrix in practice. In this paper, we give the property of equality of the two matrices by the additive property of the generalized inverse of the matrix and the zero rank of the difference between the two matrices. A new representation of generalized inverses of block matrices. By extending the correlation properties of generalized inverses, the MP-inverse expressions of 2 脳 2 block matrices with three and two zero subblocks are obtained respectively. On this basis, by using the additivity of matrix generalized inverse, the MP-inverse expressions of block matrices with and without zero subblocks are obtained. In this paper, we study the equivalent conditions between the Banachiewicz-Schur generalized inverse form of a matrix and the {1} -inverse, {1 ~ 2} -inverse, {1 ~ 3} -inverse, {1 ~ 2 ~ 2} -inverse, {1 ~ (4)} -inverse, {12 ~ (12)} -inverse of a matrix, and provide a new way of thinking for the expressions of various generalized inverses of a matrix, and extend the results to Hermit spaces. The equivalent conditions between the Banachiewicz-Schur generalized inverse form of block Hermit matrix and its various generalized inverses are obtained. Finally, Muzhang also studies the Banachiewicz-Schur weighted generalized inverse form of block matrix and its {1l0 3X} -weighted inverse, {1o 2n 3X} -weighted inverse, {1n 4Y} -weighted inverse, {124Y} -weighted inverse. The equivalent condition between. Finally, By using the idea of matrix decomposition, the representation method of MP-inverse of edge-added matrix is studied, and a new method for solving generalized inverse of general number matrix is obtained.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

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