How many eigenvalues of a random matrix are real?

Edelman, Alan; Kostlan, Eric; Shub, Michael

doi:10.1090/S0894-0347-1994-1231689-0

How many eigenvalues of a random matrix are real?
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by Alan Edelman, Eric Kostlan and Michael Shub

J. Amer. Math. Soc. 7 (1994), 247-267

DOI: https://doi.org/10.1090/S0894-0347-1994-1231689-0

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Abstract:

Let $A$ be an $n \times n$ matrix whose elements are independent random variables with standard normal distributions. As $n \to \infty$, the expected number of real eigenvalues is asymptotic to $\sqrt {2n/\pi }$. We obtain a closed form expression for the expected number of real eigenvalues for finite $n$, and a formula for the density of a real eigenvalue for finite $n$. Asymptotically, a real normalized eigenvalue $\lambda /\sqrt n$ of such a random matrix is uniformly distributed on the interval [-1, 1]. Analogous, but strikingly different, results are presented for the real generalized eigenvalues. We report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results.

References

Handbook of mathematical functions

A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR 856019

The probability that a numerical analysis problem is difficult

50

A test matrix generation suite

Alan Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9 (1988), no. 4, 543–560. MR 964668, DOI 10.1137/0609045

Eigenvalues and condition numbers of random matrices

Alan Edelman, The distribution and moments of the smallest eigenvalue of a random matrix of Wishart type, Linear Algebra Appl. 159 (1991), 55–80. MR 1133335, DOI 10.1016/0024-3795(91)90076-9
Alan Edelman, On the distribution of a scaled condition number, Math. Comp. 58 (1992), no. 197, 185–190. MR 1106966, DOI 10.1090/S0025-5718-1992-1106966-2

Random matrix eigenvalues meet numerical linear algebra

24

Bibliography of random eigenvalue literature

The circular law and the probability that a random matrix has

real eigenvalues

Jean Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Mathematical Phys. 6 (1965), 440–449. MR 173726, DOI 10.1063/1.1704292
V. L. Girko, The circular law, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 669–679 (Russian). MR 773436
V. L. Girko, Theory of random determinants, Mathematics and its Applications (Soviet Series), vol. 45, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian. MR 1080966, DOI 10.1007/978-94-009-1858-0
Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
Rameshwar D. Gupta and Donald St. P. Richards, Hypergeometric functions of scalar matrix argument are expressible in terms of classical hypergeometric functions, SIAM J. Math. Anal. 16 (1985), no. 4, 852–858. MR 793927, DOI 10.1137/0516064
Chii-Ruey Hwang, A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 145–152. MR 841088, DOI 10.1090/conm/050/841088
M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320. MR 7812, DOI 10.1090/S0002-9904-1943-07912-8
M. Kac, On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. (2) 50 (1949), 390–408. MR 30713, DOI 10.1112/plms/s2-50.5.390
Eric Kostlan, On the spectra of Gaussian matrices, Linear Algebra Appl. 162/164 (1992), 385–388. Directions in matrix theory (Auburn, AL, 1990). MR 1148410, DOI 10.1016/0024-3795(92)90386-O
E. Kostlan, On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990) Springer, New York, 1993, pp. 419–431. MR 1246137
Nils Lehmann and Hans-Jürgen Sommers, Eigenvalue statistics of random real matrices, Phys. Rev. Lett. 67 (1991), no. 8, 941–944. MR 1121461, DOI 10.1103/PhysRevLett.67.941
Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
Robb J. Muirhead, Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982. MR 652932
Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267–285. MR 1230872, DOI 10.1007/978-1-4612-2752-6_{1}9
H.-J. Sommers, A. Crisanti, H. Sompolinsky, and Y. Stein, Spectrum of large random asymmetric matrices, Phys. Rev. Lett. 60 (1988), no. 19, 1895–1898. MR 948613, DOI 10.1103/PhysRevLett.60.1895

An atlas of functions