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Eigenvalues of Doubly Stochastic Matrices, an Unfinished Story Mashreghi, Javad


According to a long standing conjecture (sometimes called the Perfect-Mirsky conjecture), the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of all regular $k$-gons with $2 \leq k\leq n$ and anchored at 1 in the closed unit disc. It is easy to verify this fact for $n =2$ and $n=3$. But, for $n\geq  4$, it was an open question at least since 1965. Mashreghi-Rivard (2007) showed that the conjecture is wrong for $n = 5$. Then Levick-Pereira-Kribs (2014) added to the mystery by showing that the conjecture is true for $n=4$. For $n \geq 6$, the loci of eigenvalues is unknown. They also came up with a new formulation of the conjecture.

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