## 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

- M. Abramowitz and I. A. Stegun,
- A. T. Bharucha-Reid and M. Sambandham,
*Random polynomials*, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR**856019**
J. W. Demmel, - 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
—, - 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
—, - 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
S. H. Lui, private communication, 1992.
- 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
J. Spanier and K. B. Oldham,

*Handbook of mathematical functions*, Dover Publications, New York, 1965.

*The probability that a numerical analysis problem is difficult*, Math. Comp.

**50**(1988), 449-480. J. W. Demmel and A. McKenney,

*A test matrix generation suite*, Argonne National Lab, MCS-P69-0389 and LAPACK working note 9. Available from netlib@na-net.ornl.gov or xnetlib.

*Eigenvalues and condition numbers of random matrices*, Ph.D. thesis, Department of Mathematics, Mass. Inst. of Technology, 1989.

*Random matrix eigenvalues meet numerical linear algebra*, SIAM News

**24**(November 1991), 11. —,

*Bibliography of random eigenvalue literature*, available electronically by anonymous FTP from math.berkeley.edu in the directory /pub/edelman. —,

*The circular law and the probability that a random matrix has*$k$

*real eigenvalues*. (submitted to J. Amer. Math. Soc.)

*An atlas of functions*, Hemisphere Publishing, Washington, 1987.

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**7**(1994), 247-267 - MSC: Primary 60F99; Secondary 15A18, 62H99
- DOI: https://doi.org/10.1090/S0894-0347-1994-1231689-0
- MathSciNet review: 1231689