The volume of the tracenonnegative polytope via the Irwin-Hall Distribution
In this work, we find an explicit expression for the volume of the trace-nonnegative polytope, the subset of Euclidean space whose coordinates lie between -1 and 1 and sum to a nonnegative number. The volume of this region provides an upper bound for the volume of a region called the realizable region, the set of vectors which can be realized as the eigenvalues of a nonnegative matrix. This region is of interest for matrix theorists working on the nonnegative inverse eigenvalue problem. To find this expression, we employ a transformation of the Irwin-Hall distribution from probability theory. We conclude by providing a general example of a non-realizable spectrum within the trace-nonnegative polytope and a characterization of the realizability of certain spectra whose entries sum to zero. The paper includes a number of open problems for further inquiry.