A characterization of trace-zero sets realizable by compensation in the SNIEP

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is so-called C-realizability, which amounts to some kind of compensation between positive and negative entries of the list. In this paper we present a combinatorial characterization of C-realizable lists with zero sum, together with explicit formulas for C-realizable lists having at most four positive entries. One of the consequences of this characterization is that the set of zero-sum C-realizable lists is shown to be a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list.