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The four-dimensional Perfect-Mirsky Conjecture


Authors: Jeremy Levick, Rajesh Pereira and David W. Kribs
Journal: Proc. Amer. Math. Soc. 143 (2015), 1951-1956
MSC (2010): Primary 15B51; Secondary 15A18, 46A55, 46H05
DOI: https://doi.org/10.1090/S0002-9939-2014-12412-9
Published electronically: December 15, 2014
MathSciNet review: 3314105
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Abstract: We verify the Perfect-Mirsky Conjecture on the structure of the set of eigenvalues for all $ n \times n$ doubly stochastic matrices in the four-dimensional case. The $ n=1,2,3$ cases have been established previously and the $ n=5$ case has been shown to be false. Our proof is direct and uses basic tools from matrix theory and functional analysis. Based on this analysis we formulate new conjectures for the general case.


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Additional Information

Jeremy Levick
Affiliation: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

Rajesh Pereira
Affiliation: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

David W. Kribs
Affiliation: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

DOI: https://doi.org/10.1090/S0002-9939-2014-12412-9
Keywords: Stochastic matrices, doubly stochastic matrices, eigenvalues, eigenvectors, linear operators.
Received by editor(s): May 29, 2013
Received by editor(s) in revised form: June 28, 2013, and November 11, 2013
Published electronically: December 15, 2014
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2014 American Mathematical Society