How many eigenvalues of a random symmetric tensor are real?

Breiding, Paul

doi:10.1090/tran/7910

How many eigenvalues of a random symmetric tensor are real?

Author: Paul Breiding
Journal: Trans. Amer. Math. Soc. 372 (2019), 7857-7887
MSC (2010): Primary 15A18, 62H99
DOI: https://doi.org/10.1090/tran/7910
Published electronically: September 6, 2019
MathSciNet review: 4029684
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article answers a question posed by Draisma and Horobet, who asked for a closed formula for the expected number of real eigenvalues of a random real symmetric tensor drawn from the Gaussian distribution relative to the Bombieri norm. This expected value is equal to the expected number of real critical points on the unit sphere of a Kostlan polynomial. We also derive an exact formula for the expected absolute value of the determinant of a matrix from the Gaussian Orthogonal Ensemble.

[1] Handbook of mathematical functions, with formulas, graphs and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966. MR 0208798
[2] Antonio Auffinger, Gérard Ben Arous, and Jiří Černý, Random matrices and complexity of spin glasses, Comm. Pure Appl. Math. 66 (2013), no. 2, 165–201. MR 2999295, https://doi.org/10.1002/cpa.21422
[3] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685
[4] Paul Breiding, The expected number of eigenvalues of a real Gaussian tensor, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 254–271. MR 3648064, https://doi.org/10.1137/16M1089769
[5] Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario, On the geometry of the set of symmetric matrices with repeated eigenvalues, Arnold Math. J. 4 (2018), no. 3-4, 423–443. MR 3949811, https://doi.org/10.1007/s40598-018-0095-0
[6] P. Breiding and S. Timme: HomotopyContinuation.jl - a package for solving systems of polynomial equations in Julia, Mathematical Software - ICMS 2018, Lecture Notes in Computer Science. Software available at www.juliahomotopycontinuation.org.
[7] Dustin Cartwright and Bernd Sturmfels, The number of eigenvalues of a tensor, Linear Algebra Appl. 438 (2013), no. 2, 942–952. MR 2996375, https://doi.org/10.1016/j.laa.2011.05.040
[8] Jan Draisma and Emil Horobeţ, The average number of critical rank-one approximations to a tensor, Linear Multilinear Algebra 64 (2016), no. 12, 2498–2518. MR 3555447, https://doi.org/10.1080/03081087.2016.1164660
[9] Alan Edelman, Eric Kostlan, and Michael Shub, How many eigenvalues of a random matrix are real?, J. Amer. Math. Soc. 7 (1994), no. 1, 247–267. MR 1231689, https://doi.org/10.1090/S0894-0347-1994-1231689-0
[10] Yan V. Fyodorov, Antonio Lerario, and Erik Lundberg, On the number of connected components of random algebraic hypersurfaces, J. Geom. Phys. 95 (2015), 1–20. MR 3357820, https://doi.org/10.1016/j.geomphys.2015.04.006
[11] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015. Translated from the Russian; Translation edited and with a preface by Daniel Zwillinger and Victor Moll; Revised from the seventh edition [MR2360010]. MR 3307944
[12] Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
[13] Robb J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. MR 652932
[14] Keith Oldham, Jan Myland, and Jerome Spanier, An atlas of functions, 2nd ed., Springer, New York, 2009. With Equator, the atlas function calculator; With 1 CD-ROM (Windows). MR 2466333
[15] The Sage Development Team, Sage mathematics software, http://www.sagemath.org.