# The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach

## TL;DR

This paper investigates the set of eigenvalues of doubly stochastic matrices, proposing a boundary conjecture, a computational method to explore this set, and evidence relating it to the Perfect-Mirsky region for certain dimensions.

## Contribution

It introduces a Boundary Conjecture for eigenvalues of doubly stochastic matrices and provides an efficient computational approach supporting this conjecture.

## Key findings

- Support for the Boundary Conjecture through computational results
- Evidence that $DS_n$ equals $PM_n$ for certain $n > 5$
- A new method for computing portions of $DS_n$ efficiently

## 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 $n \leq 4$, 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 $n > 5$.