Characterizations of *-DMP matrices over rings

Authors: YUEFENG GAO, JIANLONG CHEN

Abstract: Let $R$ be a ring with involution $*$. $R^{m\times n}$ denotes the set of all $m\times n$ matrices over $R$. In this paper, we give a characterization of the pseudo core inverse of $A\in R^{n\times n}$ in the form of $A=GDH$, $N_r(G)=0$, $N_l(H)=0$, $D^2=D=D^*$, where $N_l(A)=\{x\in R^{1\times m} | xA=0\}$ and $N_r(A)=\{x\in R^{n\times 1}~|~Ax=0\}.$ Then we obtain necessary and sufficient conditions for $A\in R^{n\times n}$, in the form of $A=GDH$, $N_r(G)=0$, $N_l(H)=0$, $D^2=D=D^*$, to be *-DMP. If $R$ is a principal ideal domain (resp. semisimple Artinian ring), then matrices expressed as that form include all $n\times n$ matrices over $R$.

Keywords: *-DMP matrix, pseudo core inverse, core-EP inverse, Drazin inverse, Moore--Penrose inverse, factorization

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