Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2

Abstract

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever ${\lambda }_2,\dots ,{\lambda }_n$ are non-positive real numbers with $1+{\lambda }_2+\dots +{\lambda }_n⩾1/2$, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely $(1,{\lambda }_2,\dots ,{\lambda }_n)$. We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.

Accepted to Linear Algebra and Its Applications, pages 12