Perron spectratopes and the real nonnegative inverse eigenvalue problem

Call an n-by-n invertible matrix S a Perron similarity if there is a real non-scalar diagonal matrix D such that SDS−1 is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra C(S):={x∈Rn:SDxS−1≥0,Dx:=diag(x)} and P(S):={x∈C(S):x1=1}, which we call the Perron spectracone and Perron spectratope, respectively. The set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest.The Perron spectracone and spectratope of Hadamard matrices are of particular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows.In addition, we provide a constructive version of a result due to Fiedler [9, Theorem 2.4] for Hadamard orders, and a constructive version of the Boyle-Handelman theorem [2, Theorem 5.1] for Suleĭmanova spectra.