The core-EP inverse: A numerical approach for its acute perturbation

This paper studies the concept of stable perturbation $B\in {\mathbb{C}}^{n\times n}$ for the core-EP inverse of a matrix $A\in {\mathbb{C}}^{n\times n}$ with index $k$. For a given stable perturbation $B$ of $A$, explicit expressions of its core-EP inverse B^{\scriptsize\textcircled{\tiny\dagger}} and its projection at zero $B^{\pi }$ are presented. Then, the perturbation bounds of \parallel B^{\scriptsize\textcircled{\tiny\dagger}}-A^{\scriptsize\textcircled{\tiny\dagger}}\parallel/\parallel A^{\scriptsize\textcircled{\tiny\dagger}}\parallel and $\parallel B^{\pi }-A^{\pi }\parallel$ are given provided that $B$ is a stable perturbation of $A$. In addition, we investigate the concept of acute perturbation of $A$. We give a perturbation analysis with respect to core-EP inverses. We provide a condition under which the acute perturbation coincides with the stable perturbation for core-EP inverses.