In this paper, the rank stability radius problem is proposed for a real matrix under structured scalar perturbations and some interesting results are achieved based on polynomial analysis. In addition, a computable formula and a two-step procedure are obtained which nicely solves the problem in this simple set up. Finally, these results on rank stability radius are used to estimate the stability robustness of descriptor systems, and for a special class of symmetric descriptor systems, the rank stability radius is proved to be equal to the system stability radius
