Algebraic Theory of Generalized Inverses

Most of the existing monographs on generalized inverses are based on linear algebra tools and geometric methods of Banach (Hilbert) spaces to introduce generalized inverses of complex matrices and operators and their related applications, or focus on generalized inverses of matrices over special rings like division rings and integral domains, and does not include the results in general algebraic structures such as arbitrary rings, semigroups and categories, which are precisely the most general cases.

In this book, five important generalized inverses are introduced in these algebraic structures. Moreover, noting that the (pseudo) core inverse was introduced in the last decade and has attracted much attention, this book also covers the very rich research results on it, so as to be a necessary supplement to the existing monographs. This book starts with decompositions of matrices, introduces the basic properties of generalized inverses of matrices, and then discusses generalizedinverses of elements in rings and semigroups, as well as morphisms in categories. The algebraic nature of generalized inverses is presented, and the behavior of generalized inverses are related to the properties of the algebraic system.

Scholars and graduate students working on the theory of rings, semigroups and generalized inverses of matrices and operators will find this book helpful.

Discusses generalized inverses in the most general setting Relates the behavior of generalized inverses to the classical ring-theoretical properties Covers results on two new types of generalized inverses

Autorentext

Jianlong Chen is a Professor at School of Mathematics, Southeast University. He is engaged in research in algebra, including ring theory, module theory, homological theory, matrix theory and generalized inverse theory.

Xiaoxiang Zhang is a Professor at School of Mathematics, Southeast University. He is engaged in research in algebra, with research interests including the theory of rings, modules, generalized inverses and homological algebra.

Zusammenfassung

"The book is self-contained and a good reference for graduate students and researchers in the field." (Mohammad Sal Moslehian, zbMATH 1542.15001, 2024)

"The book consists of eight chapters of varying lengths, four appendices, a list of references including 116 items, and a fairly detailed subject index. ... The book has been carefully edited. In many places, the key equations have been typed both in four-vector and three-vector notations. This will make the content easier to study for readers at different levels of experience in vector and tensor analysis. Another advantage is the consequent use of the International System of Units (SI)." (Radoslaw Szmytkowski, zbMATH 1541.78002, 2024)

Inhalt

Preface.- Algebraic Basic Knowledge.- Moore-Penrose Inverses.- Group Inverses.- Drazin Inverses.- Core Inverses.- Pseudo Core Inverses.- Bibliography.- Index.