Speaker
Description
Stochastic matrices are matrices with nonnegative entries whose rows each sum to $1$. When a matrix and its transpose are both stochastic, it is said to be \emph{doubly stochastic}. In 1938, Kolmogorov proposed the problem of characterizing the region of possible eigenvalues of an $n \times n$ stochastic matrix, and Karpelevich gave a complete description thirteen years later. This talk concerns the doubly stochastic analogue: characterizing the region $\omega_n$ of eigenvalues of $n \times n$ doubly stochastic matrices, which is contained in the unit disk.
Perfect and Mirsky (1965) conjectured that $\omega_n$ is the union of the regions $\Pi_k$ (the convex hulls of the $k$-th roots of unity) for $k = 1,\dots,n$. This conjecture holds for $n = 1,2,3,4$, but fails for $n = 5$. The case $n \ge 6$ remains open. In response to the scarcity of progress over the past 60 years, Levick, Pereira, and Kribs proposed a related conjecture. In this talk, I prove a stronger version of this conjecture.
The approach, based on majorization theory and geometric properties, also provides a potential general framework for characterizing $\omega_5$, as well as a numerical method for testing the conjecture for various values of $n$.