Solvability of a system of linear equations—an approach based on the generalized inverses determined by the Penrose equations ^★

Abstract

The paper aims to play an expository role, providing a tailored introduction to the theory of matrix generalized inverses determined by the so-called Penrose equations, with the Moore--Penrose inverse as the jewel in the crown. The tailoring is made taking into account applicability of the inverses to solvability of a system of linear equations, which covers, inter alia, the least squares method. The method is introduced in a formal, though compendious way, with an intention to support researchers who want to consciously utilize it in their investigations. Additionally, the paper points out various links between the generalized inverses and theory of projectors, indicating issues which are relevant from the perspective of physics. The article can be viewed as a sequel of [O.M. Baksalary and G. Trenkler, ``The Moore--Penrose inverse -- a hundred years on a frontline of physics research,'' Eur. Phys. J. H {\bf 46}, 9 (2021)], the paper prepared to celebrate the 100th anniversary of the first definition of the Moore--Penrose inverse, which shades a spotlight on the role the inverse plays in physics.