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Characterizations of the group invertibility of a matrix revisited

Tian Yongge

Demonstratio Mathematica

2022

Vol. 55, nr 1

866--890

A square complex matrix A is said to be group invertible if there exists a matrix X such that AXA=A, XAX=X, and AX=XA hold, and such a matrix X is called the group inverse of A . The group invertibility of a matrix is one of the fundamental concepts in the theory of generalized inverses, while group inverses of matrices have many essential applications in matrix theory and other disciplines. The purpose of this article is to reconsider the characterization problem of the group invertibility of a matrix, as well as the constructions of various algebraic equalities in relation to group invertible matrices. The coverage includes collecting and establishing a family of existing and new necessary and sufficient conditions for a matrix to be group invertible and giving many algebraic matrix equalities that involve Moore-Penrose inverses and group inverses of matrices through the skillful use of a series of highly selective formulas and facts about ranks, ranges, and generalized inverses of matrices, as well as block matrix operations.