The volume of the tracenonnegative polytope via the Irwin-Hall Distribution

  • Gregory Kyle Taylor
  • Pietro Paparella


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.

Author Biography

Gregory Kyle Taylor

At the time of this work, Greg Taylor was an undergraduate at The College of William & Mary. Currently, he is a graduate student at UIC. His research interests are in algebraic geometry, and his hobbies include running and playing guitar.

How to Cite
TAYLOR, Gregory Kyle; PAPARELLA, Pietro. The volume of the tracenonnegative polytope via the Irwin-Hall Distribution. Minnesota Journal of Undergraduate Mathematics, [S.l.], v. 4, n. 1, mar. 2019. ISSN 2378-5810. Available at: <>. Date accessed: 20 oct. 2020.