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 λ2,…,λn are non-positive real numbers with 1+λ2+…+λn⩾1/2, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely (1,λ2,…,λn). 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.