The probability that a random real Gaussian matrix has \(k\) real eigenvalues, related distributions, and the circular law
type: Item
instance of: scholarly article
title: The probability that a random real Gaussian matrix has \(k\) real eigenvalues, related distributions, and the circular law
zbMATH Open document ID: 0886.15024
author: Alan Edelman
published in: Journal of Multivariate Analysis
publication date: 1998-05-04T00:00:00Z
review text: The joint eigenvalue distribution of an \(n\times n\) random matrix \(A_n\) is studied. It is assumed that the entries of \(A_n\) are independent random variables identically distributed as the standard normal one. The exact form of the distribution of complex eigenvalues is found for fixed \(n\). This distribution converges as \(n\to\infty\) to the well-known circular distribution (uniform over the unit circle) derived for the first time by \textit{V. L. Girko} [Teor. Veroyatn. Primen. 29, No. 4, 669-679 (1984; Zbl 0565.60034)]. The joint distribution formula for the real Schur decomposition of \(A_n\) is obtained. As a consequence, the probability that \(A_n\) has exactly \(k\) real eigenvalues is found. In particular, the probability that all eigenvalues of \(A_n\) are real is proved to be equal to \(2^{-n(n-1)/4}\).
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1036272
zbMATH Keywords: circular law
zbMATH Keywords: eigenvalue distribution
zbMATH Keywords: random matrix
zbMATH Keywords: circular distribution
zbMATH Keywords: Schur decomposition
reviewed by: A. Khorunzhy
MaRDI profile type: Publication
full work available at URL: jmva.1996.1653
OpenAlex ID: W1999320979
DOI: 10.1006/JMVA.1996.1653
description: scientific article; zbMATH DE number 1036272