How many eigenvalues of a random matrix are real?

Abstract

Consider a random matrix whose elements are independent random variables from a standard (mean zero, variance one) normal distribution. Unless otherwise stated, we omit the distribution and simply use the term "random matrix" to denote a matrix with independent standard normally distributed elements. Other distributions are considered in §8.Here is one of our main resultsAsymptotic Number of Real Eigenvalues. If En denotes the expected number of real eigenvalues of an n-by-n random matrix, thenAsymptotic Series. As n -+ 00 ,Let A be a 50 x 50 random matrix. Figure 1 on the following page plots normalized eigenvalues AIV50 in the complex plane for fifty matrices. Thus there are 2500 dots in the figure. There are a number of striking features in the diagram. First, nearly all the normalized eigenvalues appear to fall in the interior of the unit disk. This is Girko's as yet unverified circular law [12], which states that as n gets large, AI vn is uniformly distributed in the unit disk. It follows that the proportion of eigenvalues on the real line (also strikingly visible to the eye) must tend to 0 as n -+ 00. Our results show how fast this converges. A simpler version of this circular law occurs when the random matrix has elements from a complex normal distribution; i.e. the real and imaginary parts of each element are independent standard normals. In this case the exact distribution for the eigenvalue distribution and radius can be found in Ginibre [11] and is reported by Mehta [24, p. 300] and Hwang [17]. In this case, the squares of the absolute values of the eigenvalues are independent random

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

“…The complex bulk and edge limits turn out to be the same as in the case of the much simpler complex Ginibre ensembles. The correlations of real eigenvalues in the real bulk region were obtained in [12] and the density functions were computed earlier using different techniques [8] [9]. All other results appear to be new.…”

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

“…Corollary 1 (Corollary 4.5 [15]) If λ n denotes a real eigenvalue of an n × n random matrix, then as n → ∞, the normalized eigenvalue λn √ n converges in distribution to a random variable uniformly distributed on the interval [−1, 1]…”

“…Proof: This result follows trivially from the results of Edelman [13], [15] We can, with a little work, impose a measure on the set of dynamical systems for which this result holds via results of Edelman, the neural networks [44], and some standard arguments using measure theory. Upon doing so, one nontrivial issue is understanding what such a set of dynamical system would "look" like.…”

“…Thus, convergence in distribution to uniformity on the unit disk becomes an issue. However, Edelman [15] derived a formula for the expectation value of real eigenvalues in Girko's measure in finite dimensions (we will discuss that result in section III B) which we will discuss since it is very important with respect to our results. In the process of deriving this expectation formula, Edelman also proved Girko's result.…”

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

“…Starting directly with the definition (2.1) taken at β = 1 and applying the methods of multivariate statistical analysis (Muirhead 1982), these authors have separately determined the exact mean densities † of purely real eigenvalues (Edelman et al 1994) † The two are related to the (p, q) correlation functions R…”