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

Abstract Given a square matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A , we are able to construct numerous equalities involving reasonable mixed operations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A and its conjugate transpose <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> {A}^{\ast } , Moore-Penrose inverse <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>†</m:mo> </m:mrow> </m:msup> </m:math> {A}^{\dagger } and group inverse <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>#</m:mi> </m:mrow> </m:msup> </m:math> {A}^{\#} . Such kind of equalities can be generally represented in the equation form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>†</m:mo> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>#</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> {A}^{\ast } , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>†</m:mo> </m:mrow> </m:msup> </m:math> {A}^{\dagger } , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>#</m:mi> </m:mrow> </m:msup> </m:math> {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.