Characterizations of the group invertibility of a matrix revisited

Abstract

A square complex matrix AA is said to be group invertible if there exists a matrix XX such that AXA=AAXA=A, XAX=XXAX=X, and AX=XAAX=XA hold, and such a matrix XX is called the group inverse of AA. 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.

Keywords

• block matrix
• group inverse
• moore-penrose inverse
• range
• rank
• 15a03
• 15a09
• 15a24