Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology

Abstract

Given a square matrix AA, we are able to construct numerous equalities involving reasonable mixed operations of AA and its conjugate transpose A∗{A}^{\ast }, Moore-Penrose inverse A†{A}^{\dagger } and group inverse A#{A}^{\#}. Such kind of equalities can be generally represented in the equation form f(A,A∗,A†,A#)=0f\left(A,{A}^{\ast },{A}^{\dagger },{A}^{\#})=0. In this article, the author constructs a series of simple or complicated matrix equalities composed of AA, A∗{A}^{\ast }, A†{A}^{\dagger }, A#{A}^{\#} and their algebraic operations, as well as established various explicit formulas for calculating the ranks of these matrix expressions. Many applications of these matrix rank equalities are presented, including a broad range of necessary and sufficient conditions for a square matrix to be range-Hermitian and Hermitian/skew-Hermitian, respectively.

Keywords

• block matrix
• group inverse
• moore-penrose inverse
• range
• rank
• reverse-order law
• 15a09
• 15a24