Various solutions to reverse order law problems
Dragana S. Cvetković‐Ilić, Marko S. Djikić · Linear and Multilinear Algebra · 2015
In this paper, we consider the reverse order law for -inverses of matrices and we do that taking two completely different approaches. The paper is an illustration of how in working with problems related to generalized inverses of matrices, one can sometimes get around a plethora of complicated formulas concerning maximal and/or minimal ranks of various expressions, and give purely algebraic proofs of the same assertions or take a different path altogether and go deeper into the essence of the definition of generalized inverses. We also give a constructive solution to a reverse order law problem on Banach spaces and its application in correction of ‘The ghost of an index theorem’ given by Harte.