The weak perturbation of matrices for the Moore-Penrose inverse in rings with involution

Abstract

Let $R$ be a $\ast$-ring. For any $T\in R_{m\times n}$, $E\in R_{m\times m}$ and $F\in R_{n\times n}$, a matrix $M$ over a $*$-ring is called a multiplicative perturbation of $T$ if $M=ETF^*$. The initial goal of this paper is to introduce a type of multiplicative perturbation, called weak perturbation. Suppose that $T$, $L_E$ and $R_F$ have Moore-Penrose inverses $T^\dagger$, $L_E^\dagger$ and $R_F^\dagger$, respectively. Then, a multiplicative perturbation $ETF^*$ of $T$ is said to be a weak perturbation if $L_E^\dagger L_ET=TR_F^\dagger R_F$, where $L_E=ETT^\dagger+I_m-TT^\dagger$ and $R_F=FT^\dagger T+I_n-T^\dagger T$. It is worth noting that the definition of weak perturbation for the Moore-Penrose inverse of matrices over rings only satisfies one of the equations in the case of a complex matrix. It is shown that if $M$ is a weak perturbation of $T$, then $M$ has a Moore-Penrose inverse if and only if the Moore-Penrose inverses of $ETT^\dagger$ and $T^\dagger TF^*$ both exist. Moreover, $M^\dagger=(T^\dagger TF^*)^\dagger T^\dagger (ETT^\dagger)^\dagger.$ Necessary and sufficient conditions for the existence as well as the expressions for $M^\dagger$ are derived under the condition that $M$ is the weak perturbation of $T$. If $M$ is a multiplicative perturbation of $T$, and $(ETT^\dagger)^\dagger$ and $(T^\dagger TF^*)^\dagger$ both exist, inspired by an alternative expression for $M$ as $M=(ETT^\dagger)\cdot T\cdot(T^\dagger TF^*)$, we prove that $M^\dagger$ exists if and only if $S^\dagger_{L,N^{-1}}$ exists, where $S^\dagger_{LN^{-1}}$ is the weighted Moore-Penrose inverse for certain matrices $S$, $L$ and $N$ associated to the triple $(T, E, F)$. Moreover, $M^\dagger=(T^\dagger TF^*)^\dagger S^\dagger_{L,N^{-1}}(ETT^\dagger)^\dagger$.