Perturbation analysis of tensor $(\mathcal{B},\mathcal{C})$-inverse via Einstein product

TL;DR

This paper studies how small perturbations in a tensor affect its generalized inverses, focusing on the $( ext{B,C})$-inverse via Einstein product, and provides insights into the stability of these inverses under perturbations.

Contribution

It offers a detailed analysis of the perturbation effects on tensor $( ext{B,C})$-inverses, extending inverse stability theory to tensor algebra with Einstein product.

Findings

Derived bounds for the change in tensor inverses under perturbations

Established conditions for the stability of tensor $( ext{B,C})$-inverses

Analyzed the relationship between original and perturbed tensor inverses

Abstract

We investigate the influence of a relatively small perturbation on various generalized inverses functions or quantities derived from a tensor $\mathcal{A}$.When a small tensor perturbation \(\mathcal{E}\) is introduced, it becomes challenging to analyze generalized inverses of the perturbed tensor \( \mathcal{D} =\mathcal{A}+\mathcal{E}\) and to determine how this perturbation affects a generalized inverse of $\mathcal{A}$.Our main goal is to understand the relationship between $\mathcal{D}^\Game$ and \( \mathcal{A}^\Game \), where $(\cdot)^\Game$ denotes a specific generalized inverse or a class of generalized inverses.In particular, classes of tensor inner, outer, and $(\mathcal{B},\mathcal{C})$ inverses are considered.