On the structure of the set of symmetric matrices with the Perron-Frobenius property

In this paper we explore the geometry of the set of symmetric Perron-Frobenius matrices, whose spectral radius is an eigenvalue with a nonnegative associated eigenvector. We prove that the set, which is not convex, is star convex with the identity matrix as the center. We also describe a few aspects of the geometric structure of the set.