The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach

The problem of determining DSn, the complex numbers that occur as an eigenvalue of an n-by-n doubly stochastic matrix, has been a target of study for some time. The Perfect-Mirsky region, PMn, is contained in DSn and is known to be exactly DSn for but strictly contained within DSn for n = 5. Here, we present a Boundary Conjecture that asserts that the boundary of DSn is achieved by eigenvalues of convex combinations of pairs of (or single) permutation matrices. We present a method to efficiently compute a portion of DSn and obtain computational results that support the Boundary Conjecture. We also give evidence that DSn is equal to PMn for certain n > 5.