Additive properties and absorption laws for generalized inverses

TL;DR

This paper investigates the conditions under which absorption and additive laws hold for pseudo core inverses in rings, providing characterizations and extending results to complex matrices.

Contribution

It establishes necessary and sufficient conditions for absorption and additive properties of pseudo core inverses in rings, including matrix cases, expanding understanding of generalized inverse behaviors.

Findings

Absorption law holds iff 1 + a^{D}b is invertible.

Additive property characterized by (1 + a^{D}b)^{-1}a^{D}.

Results extended to complex matrices.

Abstract

Let $a,~f$ be elements in a ring with pseudo core inverses $a^{\scriptsize\textcircled{\tiny D}}$, $f^{\scriptsize\textcircled{\tiny D}}$, and let $b=f-a$. We prove that the absorption law $a^{\scriptsize\textcircled{\tiny D}}(a+f)f^{\scriptsize\textcircled{\tiny D}}=a^{\scriptsize\textcircled{\tiny D}}+f^{\scriptsize\textcircled{\tiny D}}$ holds if and only if $1+a^{\scriptsize\textcircled{\tiny D}}b$ is invertible and the additive property $f^{\scriptsize\textcircled{\tiny D}}=(1+a^{\scriptsize\textcircled{\tiny D}}b)^{-1}a^{\scriptsize\textcircled{\tiny D}}$ is satisfied. We further characterize these properties and establish analogous results for other generalized inverses. Finally, we apply these results to the case of complex matrices.