广义逆的代数理论-代数基本知识,Moore-Penrose逆

ALGEBRAIC THEORY OF GENERALIZED INVERSES:ALGEBRAIC BASIC KNOWLEDGE AND MOORE-PENROSE INVERSES

本文是《广义逆的代数理论》专著的第一部分,包括"代数基本知识"和"Moore-Penrose逆"两个章节的内容.第一章"代数基本知识"(参阅本文第1-6节)介绍了本书所涉及的基本知识,为后续研究奠定基础:我们首先简要回顾了复矩阵的一些分解,然后介绍一些关于环和对合的基本知识,例如环的FP-内射性、正则性、*-正则性和有限性.在第二章"Moore-Penrose逆"中(参阅本文第7-14节),我们给出了复矩阵的Moore-Penrose逆的一些计算方法,并研究了*-环中MoorePenrose逆的特征,研究了乘积、分块矩阵、友矩阵、投影元的差(乘积)以及态射的和的MoorePenrose可逆性.此外,我们还研究了针对Moore-Penrose逆的J acobson引理.

This is the first part of a series of Algebraic Theory of Generalized Inverses,which includes two chapters:Algebraic Basic Knowledge and Moore-Penrose Inverses.The chapter Algebraic Basic Knowledge(see our Sec.1-6)is devoted to set the stage for the whole discussion.We first review some decompositions of complex matrices briefly,and then collect some of the basic knowledge of rings and involutions such as FP-injectivity,regularity,*-regularity and finiteness of rings.In the chapter MoorePenrose Inverses(see our Sec.7-14),we give some calculation methods for MoorePenrose inverses of complex matrices and study characterizations of Moore-Penrose inverses in a*-ring(*-semigroup);Moore-Penrose invertibility of a product,block matrix,companion matrix,difference(product)of projections and sum of morphisms.Among others,we also investigate Jacobson’s Lemma for Moore-Penrose inverses.