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

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.