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

TL;DR

This paper develops a numerical approach to analyze the stability of the core-EP inverse of matrices under perturbations, providing explicit formulas, bounds, and conditions for acute perturbations.

Contribution

It introduces explicit expressions and bounds for the core-EP inverse under stable perturbations and explores the concept of acute perturbations in this context.

Findings

Explicit formulas for perturbed core-EP inverse and projections.

Perturbation bounds for the core-EP inverse and projections.

Conditions under which acute perturbation equals stable perturbation.

Abstract

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.