Papers › The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach
The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach
Amit Harlev, Charles R. Johnson, Derek Lim
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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 n ≤4, 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.
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