Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are to be bounded below by 1/2
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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 λ_2, ..., λ_n are non-positive real numbers with 1 + λ_2 + ... + λ_n ≥ 1/2, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely (1, λ_2, ..., λ_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 Nadar et al. We also provide some examples and applications of our results.
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