Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem

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 $SD{S}^{-1}$ is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra $\mathcal{C}(S):=\{x\in {\mathbb{R}}^n:S{D}_x{S}^{-1}\ge 0,D_x:=\text{diag}(x)\}$ 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\uımanova spectra.

To appear in Linear Algebra and its Applications

Cited by in corpus (6)

• Row Cones, Perron Similarities, and Nonnegative Spectra
• Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2
• On the realizability of the critical points of a realizable list
• Perron similarities and the nonnegative inverse eigenvalue problem
• Character tables are ideal Perron similarities
• The Volume of the Trace-Nonnegative Polytope via the Irwin-Hall Distribution