广义逆的代数理论——群逆,Drazin逆

ALGEBRAIC THEORY OF GENERALIZED INVERSES:GROUP INVERSES AND DRAZIN INVERSES

本文是《广义逆的代数理论》专著的第二部分,包括"群逆"和"Drazin逆"两个章节的内容.第一章"群逆"(参阅本文第1-9节)首先给出了复矩阵群逆存在的充分必要条件,然后研究了半群或环中群逆的刻画,研究了乘积paq、态射的和、两个群可逆元素的和、乘积、分块矩阵以及友矩阵的群可逆性.在第二章"Drazin逆"中(参阅本文第10-17节),我们给出了复矩阵的Drazin逆的一些计算方法,并研究了半群或环中Drazin逆的特征,研究了元素乘积、投影元的差与乘积、环上矩阵、态射的和的Drazin可逆性.此外,我们还研究了针对Drazin逆的加法性质和Jacobson引理.

This is the second part of a series of Algebraic Theory of Generalized Inverses,which includes two chapters:Group Inverses and Drazin Inverses.The chapter Group Inverses(see our Sec.1-9)first gives several sufficient and necessary conditions which guarantee the existence of the group inverse of a complex matrix,and then investigates the characterizations of group inverses in semigroups or rings;group invertibility of product paq,a sum of morphisms,the sum of two group invertible elements,a product,block matrices and a companion matrix.In the chapter Drazin Inverses(see our Sec.10-17),we give some calculation methods for Drazin inverses of complex matrices and study characterizations of Drazin inverses in semigroups or rings.Moreover,we also investigate Drazin invertibility of the product and difference of idempotents,matrices over a ring and a sum of morphisms.Among others,additive properties and Jacobson’s Lemma for Drazin inverses are presented.