Abstract

The problem of determining DS_n, 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, PM_n, is contained in DS_n and is known to be exactly DS_n for $n\le 4,$ but strictly contained within DS_n for n = 5. Here, we present a Boundary Conjecture that asserts that the boundary of DS_n is achieved by eigenvalues of convex combinations of pairs of (or single) permutation matrices. We present a method to efficiently compute a portion of DS_n and obtain computational results that support the Boundary Conjecture. We also give evidence that DS_n is equal to PM_n for certain n > 5.

摘要

确定DS _n的问题，即作为n × n双随机矩阵的特征值出现的复数，一直是研究的目标。Perfect-Mirsky 区域PM _n包含在DS _n中，并且已知恰好是DS _n$n\le 4,$但在n = 5 时严格包含在DS _n内 。在这里，我们提出一个边界猜想，断言DS _n的边界是通过成对（或单个）置换矩阵的凸组合的特征值实现的。我们提出了一种有效计算DS _n的一部分并获得支持边界猜想的计算结果的方法。我们还提供证据证明DS _n等于PM _n对于某些n  > 5。