The new revisitation of core EP inverse of matrices

Abstract

In this paper, we study some properties of \emph{core-EP inverse} of square matrices. Firstly, we extend the theorem proved by K.M. Prasad and K.S. Mohana obtained. Then some properties of core-EP inverse have been given, through applying the conditions $(AX)^{*}=AX$, $XA^{k+1}=A^{k}$ and $\mathcal{R}(X)=\mathcal{R}(X^{*})=\mathcal{R}(A^{k})$. Secondly, we get some characterizations of \emph{core-EP inverse} by employing the conditions $AX=P_{A^{k}}$ and $XA=P_{\mathcal{R}(A^{k}),\mathcal{N}((A^{k+1})^{*}A)}$. Finally, we get some properties of \emph{core-EP inverse} by utilizing the condition $A^{k+1}X=A^{k}P_{A^{k}}$.