The four-dimensional Perfect-Mirsky Conjecture
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- by Jeremy Levick, Rajesh Pereira and David W. Kribs PDF
- Proc. Amer. Math. Soc. 143 (2015), 1951-1956 Request permission
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.References
- Luca Benvenuti and Lorenzo Farina, Eigenvalue regions for positive systems, Systems Control Lett. 51 (2004), no. 3-4, 325–330. MR 2037263, DOI 10.1016/j.sysconle.2003.09.009
- Garrett Birkhoff, Three observations on linear algebra, Univ. Nac. Tucumán. Revista A. 5 (1946), 147–151 (Spanish). MR 0020547
- N. Dmitriev and E. Dynkin, On characteristic roots of stochastic matrices, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 10 (1946), 167–184 (Russian, with English summary). MR 0017269
- Charles R. Johnson, An inclusion region for the field of values of a doubly stochastic matrix based on its graph, Aequationes Math. 17 (1978), no. 2-3, 305–310. MR 491769, DOI 10.1007/BF01818568
- Charles R. Johnson, Row stochastic matrices similar to doubly stochastic matrices, Linear and Multilinear Algebra 10 (1981), no. 2, 113–130. MR 618581, DOI 10.1080/03081088108817402
- F. I. Karpelevič, On characteristic roots of matrices with nonnegative coefficients, Uspehi Matem. Nauk (N.S.) 4 (1949), no. 5(33), 177–178. MR 0031462
- Javad Mashreghi and Roland Rivard, On a conjecture about the eigenvalues of doubly stochastic matrices, Linear Multilinear Algebra 55 (2007), no. 5, 491–498. MR 2363549, DOI 10.1080/03081080600899296
- Henryk Minc, Nonnegative matrices, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 932967
- Hazel Perfect and L. Mirsky, Spectral properties of doubly-stochastic matrices, Monatsh. Math. 69 (1965), 35–57. MR 175917, DOI 10.1007/BF01313442
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
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3314105