On a generalization of the Johnson-Newman theorem to multiple rank-one perturbations

TL;DR

This paper simplifies and extends the Johnson-Newman theorem to apply to multiple arbitrary rank-one perturbations, providing a more accessible and general criterion for matrix similarity.

Contribution

It offers a simplified proof and an improved, more general condition for the theorem's applicability to multiple rank-one perturbations.

Findings

Simplified proof of the generalized Johnson-Newman theorem

Extended the theorem to arbitrary rank-one perturbations

Provided a more straightforward condition for matrix similarity

Abstract

Wang and Zhao (Adv. Appl. Math. 173 (2026) 102994) generalized the classic Johnson-Newman theorem on simultaneous similarity of symmetric matrices from a single rank-one perturbation to multiple rank-one perturbations. However, their result applies only to specific rank-one perturbations, and the given condition is quite involved as it relies on multivariate polynomials. We provide a simple proof of their result, leading to an improved version with a simplified condition that holds for arbitrary rank-one perturbations.