When is $(A+B)^{\dagger}=A^{\dagger}+B^{\dagger}$?

TL;DR

This paper investigates the specific conditions under which the Moore-Penrose inverse of a sum of matrices equals the sum of their Moore-Penrose inverses, also exploring similar conditions for the group inverse.

Contribution

It provides new necessary and sufficient conditions for the equality $(A+B)^{ ext{dagger}}=A^{ ext{dagger}}+B^{ ext{dagger}}$ and extends the analysis to the group inverse.

Findings

Derived conditions for the Moore-Penrose inverse equality

Extended analysis to the group inverse case

Identified scenarios where the inverse sum property holds

Abstract

We address the question as to when it is true that $(A+B)^{\dagger}=A^{\dagger}+B^{\dagger},$ where $\dagger$ denotes the Moore-Penrose inverse. A similar question is addressed for the group inverse.