Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
Yongge Tian · Demonstratio Mathematica · 2025
Abstract Given a square matrix A A , we are able to construct numerous equalities involving reasonable mixed operations of A A 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 # ) = 0 f\left(A,{A}^{\ast },{A}^{\dagger },{A}^{\#})=0 . In this article, the author constructs a series of simple or complicated matrix equalities composed of A A , 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.