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The paper considers various problems concerned with the Moore–Penrose inverse of a sum of two matrices. By establishing several original results and by combining various facts known in the literature, the article reveals a number of emerging features of the inverse. The investigations shed also a spotlight on different proving approaches useful to cope with the problems in question. Rather than focusing exclusively on the main topic, the considerations endeavour to place it in a wider context, linking it with different matrix notions, facts, and tools, as well as indicating areas of its applications originating from, e.g. computational methods, physics, statistics, discriminant analysis, or neuroscience. [ABSTRACT FROM AUTHOR]

The paper investigates the class of square matrices which have idempotent Moore–Penrose inverse. A number of original characteristics of the class are derived and new properties identified. Additionally, some isolated results scattered in the literature are recaptured and rederived in an extended or generalized form. The research was inspired by some matrix equations (occurring, e.g. in physics) which either directly involve matrices with an idempotent Moore–Penrose inverse or whose solvability under the assumption that the matrices involved have an idempotent Moore–Penrose inverse provides an emerging insight into the corresponding theory. [ABSTRACT FROM AUTHOR]