Row Cones, Perron Similarities, and Nonnegative Spectra

TL;DR

This paper explores the relationship between specific matrix cones related to the real nonnegative inverse eigenvalue problem, introducing new concepts like row Hadamard conic matrices and characterizing conditions for cone equality.

Contribution

It introduces the concept of row Hadamard conic matrices and characterizes when the row cone equals the spectracone in the context of Perron similarities.

Findings

Characterizations of when _r(S) = (S)

Introduction of row Hadamard conic matrices

Explicit examples of cone relationships

Abstract

In further pursuit of the diagonalizable \emph{real nonnegative inverse eigenvalue problem} (RNIEP), we study the relationship between the \emph{row cone} $\mathcal{C}_r(S)$ and the \emph{spectracone} $\mathcal{C}(S)$ of a Perron similarity $S$. In the process, a new kind of matrix, \emph{row Hadamard conic} (RHC), is defined and related to the D-RNIEP. Characterizations are given when $\mathcal{C}_r(S) = \mathcal{C}(S)$, and explicit examples are given for all possible set-theoretic relationships between the two cones. The symmetric NIEP is the special case of the D-RNIEP in which the Perron similarity $S$ is also orthogonal.