Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem

TL;DR

This paper introduces a recursive method to characterize all symmetrically realisable spectra for nonnegative symmetric matrices, unifying and extending over forty years of sufficient conditions in the field.

Contribution

It presents a new recursive construction method for symmetrically realisable lists, linking and generalizing existing sufficient conditions for the SNIEP.

Findings

The recursive method captures all known sufficient conditions.

It unifies previous criteria into a single framework.

The approach advances understanding of the SNIEP structure.

Abstract

We say that a list of real numbers is "symmetrically realisable" if it is the spectrum of some (entrywise) nonnegative symmetric matrix. The Symmetric Nonnegative Inverse Eigenvalue Problem (SNIEP) is the problem of characterising all symmetrically realisable lists. In this paper, we present a recursive method for constructing symmetrically realisable lists. The properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years, starting with the work of Fiedler in 1974. We show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing.