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A matricial view of the Karpelevi Theorem.pdf

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A matricial view of the Karpelevi Theorem.pdf

Linear Algebra and its Applications 520 (2017) 1–15 Contents lists available at ScienceDirect Linear Algebra and its Applications /locate/laa A matricial view of the Karpelevi? Theorem Charles R. Johnson a, Pietro Paparella b,? a Department of Mathematics, College of William Mary, Williamsburg, VA 23187-8795, USA b Division of Engineering and Mathematics, University of Washington Bothell, Bothell, WA 98011-8246, USA article info Article history: Received 2 December 2016 Accepted 11 January 2017 Available online 16 January 2017 Submitted by R. Brualdi MSC: 15A18 15A29 15B51 Keywords: Stochastic matrix Doubly stochastic matrix Karpelevi? arc Karpelevi? region Ito polynomial Realizing matrix abstract The question of the exact region in the complex plane of the possible single eigenvalues of all n-by-n stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevi? in 1951 after a partial result by Dmitriev and Dynkin in 1946. The Karpelevi? result is unwieldy, but a simpli?cation was given by Dokovi? in 1990 and Ito in 1997. The Karpelevi? region is determined by a set of boundary arcs each connecting consecutive roots of unity of order less than n. It is shown here that each of these arcs is realized by a single, somewhat simple, parameterized stochastic matrix. Other observations are made about the nature of the arcs and several further questions are raised. The doubly stochastic analog of the Karpelevi? region remains open, but a conjecture about it is ampli?ed. ? 2017 Elsevier Inc. All rights reserved. * Corresponding author. E-mail addresses: crjohn@ (C.R. Johnson), pietrop@ (P. Paparella). URL: /pietrop/ (P. Paparella). /10.1016/j.laa.2017.01.009 0024-3795/? 2017 Elsevier Inc. All rights reserved. 2 C.R. Johnson, P. Paparella / Linear Algebra and its Applications 520 (2017) 1–15 1. Introduction In [11], Kolmogorov posed the problem of characterizing the subset of the complex plane, denoted by Θn, that consists of the individual eigenvalues of all