Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem

TL;DR

This paper introduces Perron spectratopes, characterizes Perron similarities, and explores their role in the real nonnegative inverse eigenvalue problem, with particular focus on Hadamard matrices and spectra.

Contribution

It provides new characterizations of Perron similarities, studies Perron spectratopes, and offers constructive methods for specific spectral problems related to Hadamard matrices.

Findings

Perron spectratopes can cover all normalized real spectra of diagonalizable nonnegative matrices.

For Hadamard matrices, Perron spectratopes coincide with convex hulls of rows.

Constructive versions of classical theorems are provided for Hadamard orders and Sulemanova spectra.

Abstract

Call an $n$-by-$n$ invertible matrix $S$ a \emph{Perron similarity} if there is a real non-scalar diagonal matrix $D$ such that $S D S^{-1}$ is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra $\mathcal{C}(S) := \{ x \in \mathbb{R}^n: S D_x S^{-1} \geq 0,~D_x := \text{diag}(x) \}$ and $\mathcal{P})(S) := \{x \in \mathcal{C}(S) : x_1 = 1 \}$, which we call the \emph{Perron spectracone} and \emph{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 (\cite[Theorem 2.4]{f1974}) for Hadamard orders, and a constructive version of \cite[Theorem 5.1]{bh1991} for Sule\u{\i}manova spectra.