Perron spectratopes and the real nonnegative inverse eigenvalue problem

Abstract

Call an n-by-n invertible matrix S a Perron similarity if there is a real nonscalar diagonal matrix D such that SDS −1 is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra C (S) :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 [2, Theorem 5.1] for Suleȋmanova spectra.

Realizing Suleĭmanova spectra via permutative matrices, II

“…Detracting from both demonstrations is that they are not constructive, i.e., a realizing matrix is not explicitly given. Johnson and Paparella [8] give a constructive proof that every Suleimanova spectrum is symmetrically realizable via similarity by Hadamard matrices, which we will show gives a constructive proof for lists satisfying (6) and (7). A positive integer n is called a Hadamard order if there is a Hadamard matrix of order n. It is well-known, and otherwise easy to establish, that if n is a Hadamard order greater than one, then n can not be odd.…”

“…, λ n } is called normalized if λ 1 = 1 ≥ λ 2 ≥ · · · ≥ λ n . Johnson and Paparella [8] used Hadamard matrices to establish the following result (recall that a nonnegative matrix A is called doubly stochastic if every row and column sum to unity). Theorem 6.5 ([8, Theorem 6.3]).…”

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

“…. Hence the two problems of D-RNIEP and SNIEP are di erent at least in this case (see also the recent work on the diagonalizable real nonnegative inverse eigenvalue problem in [10], [11], [12] and [22]). …”

On the realizability of the critical points of a realizable list

“…is called the spectracone of S [18,19]. It is known that the spectracones corresponding to Hadamard matrices comprise a large class of realizable real spectra [19].…”

“…is called the spectracone of S [18,19]. It is known that the spectracones corresponding to Hadamard matrices comprise a large class of realizable real spectra [19]. Furthermore, spectracones corresponding to DFT matrices comprise a large class of realizable nonreal spectra [18].…”