相对于两个元素的广义逆研究

In 2012, Drazin first introduced the definition of the generalized inverse related to two elements---(b, c)-inverse, which unified the classical generalized inverses, such as Moore-Penrose inverse, Drazin inverse, Mary inverse and core inverse. This project is based on special (b,c)-inverses, and investigate related properties of the (b, c)-inverse systematically. We will discuss several aspects that contained the existences and the expressions of the (b, c)-inverse, the (b, c)-invertibility of the sum and product of elements and the generalizations of the (b, c)-inverse. In the research process, some characterizations of the (b, c)-inverse and related generalized inverses are given in terms of the invertible elements. Moreover, using the methods of the decomposition of matrices and the similarity transformation of matrices, the (b, c)-invertibility of block matrices are investigated. By introducing the partial orders of the (b, c)-inverse, the (b, c)-invertibility of the sum and product of elements will be considered. Moreover, the one-sided core inverses will be studied further. Base on the regularity of elements, we investigate strongly one-sided (b, c)-inverses, and the related results on one-sided generalized inverses are generalized. Because it has the generality for the (b, c)-inverse, the researchs on the (b, c)-inverse will provide a new platform for the further studies on the classical generalized inverse. Using the partial order of generalized inverses and one-side generalized inverses as tools will provide new power for researches on the (b, c)-inverse. The results obtained in this project are expected to deepen understanding of the (b, c)-inverse and related generalized inverses.

2012年Drazin首次引入相对于两个元素的广义逆---(b,c)-逆，统一了Moore-Penrose逆、Drazin逆、Mary逆与core逆等经典广义逆。本项目以特殊(b,c)-逆为基础，系统研究(b,c)-逆的相关性质，着重探讨(b,c)-逆的存在性与显式表示、元素和与积的(b,c)可逆性及广义(b,c)逆。我们拟利用可逆元给出(b,c)-逆及相关广义逆的刻画；同时借助矩阵分解和相似变换来研究分块矩阵的(b,c)-可逆性；引入(b,c)-偏序，探讨元素和与积的(b,c)-可逆性；讨论单边core逆，通过元素正则性，研究强单边(b,c)-逆，进而推广单边广义逆的相关结果。由于(b,c)-逆具有一般性，对(b,c)-逆的深入研究将为经典广义逆的发展提供新的平台，而偏序条件与单边广义逆又为(b,c)-逆的研究注入新的活力。本项目预期成果有望深化人们对(b,c)-逆及相关广义逆的认识。

2012年Drazin首次引入相对于两个元素的广义逆---(b,c)-逆，统一了Moore-Penrose逆、Drazin逆、Mary逆与core逆等经典广义逆。本项目以特殊(b,c)-逆为基础，系统研究(b,c)-逆及相关广义逆的性质，着重探讨(b,c)-逆的存在性与显式表示、讨论单边core逆和单边Mary逆的相关刻画，通过元素正则性，研究强单边(b,c)-逆，进而推广单边广义逆的相关结果。讨论了(b,c)-逆的EP性质和吸收律问题。由于(b,c)-逆具有一般性，对(b,c)-逆的深入研究将为经典广义逆的发展提供新的平台，而偏序条件与单边广义逆又为(b,c)-逆的研究注入新的活力。本项目的相关成果深化了人们对(b,c)-逆及相关广义逆的认识。