The symmetric doubly stochastic inverse spectral problem is the problem of de-termining necessary and sufficient conditions for a real n-tuple to be the spectrum of an n × n symmetric doubly stochastic matrix. For n ≥ 4, this problem remains open though many partial results are known. In this note, we present a new family of necessary conditions for this problem using some matrix trace inequalities. In addition, we prove that this family of new inequalities sharpen the existing known necessary conditions for the inverse spectral problem of nonnegative matrices. Finally, we prove that these necessary conditions are not sufficient for the case n = 3.

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