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Invertibility of structured perturbations of singular matrices over unital rings
Abstract
In this article, we study the invertibility of matrices of the form A+EDF, where A and D are square matrices over a unital, not necessarily commutative, ring and A is singular. Under natural hypotheses on kernels and images, we prove that A+EDF is invertible if and only if D is invertible, and obtain an explicit formula for the inverse of A+EDF by purely algebraic methods. When the coefficient ring is stably finite, or more specifically a field, we obtain an invertibility criterion under weaker hypotheses. Our results extend and sharpen a theorem of Eriksson and Nordqvist for complex matrices.
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BibTeX
@misc{oinert2026invertibility,
title = {{Invertibility of structured perturbations of singular matrices over unital rings}},
author = {Johan Öinert},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01948},
eprint = {2610.01948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01948},
}
RIS
TY - UNPB
TI - Invertibility of structured perturbations of singular matrices over unital rings
AU - Johan Öinert
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01948
UR - https://arxiv.org/abs/2610.01948
AB - In this article, we study the invertibility of matrices of the form A+EDF, where A and D are square matrices over a unital, not necessarily commutative, ring and A is singular. Under natural hypotheses on kernels and images, we prove that A+EDF is invertible if and only if D is invertible, and obtain an explicit formula for the inverse of A+EDF by purely algebraic methods. When the coefficient ring is stably finite, or more specifically a field, we obtain an invertibility criterion under weaker hypotheses. Our results extend and sharpen a theorem of Eriksson and Nordqvist for complex matrices.
ER -