本文選題：分塊矩陣廣義逆　切入點(diǎn)：Banachiewicz-Schur形式　出處：《北京交通大學(xué)》2017年碩士論文

【摘要】：矩陣的廣義逆理論一直都是世界矩陣論領(lǐng)域中一個(gè)非常重要的討論分支,并且在工程運(yùn)算求解線性方程組的一般解、最小二乘解以及最優(yōu)化控制等研究中,廣義逆理論都起著不可忽視的作用。對(duì)矩陣廣義逆的研究,我們通常采用將矩陣分塊成2 × 2的分塊矩陣思想,通過研究其四個(gè)子塊得到原矩陣的廣義逆的相關(guān)性質(zhì)。經(jīng)過學(xué)者的廣泛研究,得到的分塊矩陣廣義逆的表達(dá)式形式多樣,但當(dāng)運(yùn)用到實(shí)際計(jì)算一般數(shù)陣的MP-逆問題上仍然具有很大的困難。本文將通過矩陣廣義逆的可加性及兩矩陣差的秩為零則這兩個(gè)矩陣相等的性質(zhì)給出了2 × 2分塊矩陣廣義逆的新表示方法。首先,通過推廣廣義逆的相關(guān)性質(zhì)分別得到帶有三個(gè)和帶有兩個(gè)零子塊的2 × 2分塊矩陣MP-逆表達(dá)式,在此基礎(chǔ)上采用矩陣廣義逆的可加性得到帶有一個(gè)零子塊及不帶有零子塊的分塊矩陣MP-逆表達(dá)式。其次,研究了矩陣的Banachiewicz-Schur廣義逆形式與矩陣{1}-逆、{1,2}-逆、{1,3}-逆、{1,2,3}-逆、{1,4}-逆,{12,4}-逆之間的等價(jià)條件,為矩陣的各種廣義逆的表達(dá)式提供了一種新的思路。并將所得結(jié)論推廣到了Hermit空間,得到了分塊Hermit矩陣的Banachiewicz-Schur廣義逆形式與其各種廣義逆之間的等價(jià)條件。木章最后還研究了分塊矩陣的Banachiewicz-Schur加權(quán)廣義逆形式與其{1,3X}-加權(quán)逆、{1,2,3X}-加權(quán)逆、{1,4Y}-加權(quán)逆、{12,4Y}-加權(quán)逆之間的等價(jià)條件。最后,采用矩陣分解思想,研究了加邊矩陣的MP-逆的表示方法,由此可以得到一種新的求解一般數(shù)陣廣義逆的方法。
[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.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O151.21

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本文編號(hào)：1699854