On the structural eigenvalues of block random matrices
type: Item
instance of: scholarly article
title: On the structural eigenvalues of block random matrices
zbMATH Open document ID: 0862.15016
DOI: 10.1016/0024-3795(94)00356-4
author: Ferenc Juhász
published in: Linear Algebra and its Applications
publication date: 1997-01-29T00:00:00Z
review text: The paper concerns the eigenvalues of an \(n\times n\) random matrix \(A_n=(a_{ij})\) consisting of \(m^2\) square blocks. The entries are such that \(a_{ij}-\mathbb E a_{ij}\) are independent identically distributed random variables with variance \(\sigma^2\) while the averages \(\mathbb E a_{ij}\) differ from one block to another and are equal for all entries lying inside a block. The main result is that in the limit of infinite \(n\), \(A_n\) has \(m\) eigenvalues that are of order \(n\); the magnitude of these is determined by the values of \(\mathbb E a_{ij}\). The rest eigenvalues are of order \(\sqrt n\) with magnitude proportional to \(\sigma\). The proof is based on a combination of perturbation theorems for eigenvalues of matrices with diagonal Jordan normal form and the fundamental fact of random matrix spectral theory that for \(\widehat A_n=A_n-\mathbb E A_n\) one has \[ |\widehat A_n|= 2\sigma\sqrt n (1+o(1)),\quad n\to\infty \] [see \textit{Z. D. Bai} and \textit{Y. Q. Yin}, Probab. Theory Relat. Fields 73, 555--569 (1986; Zbl 0586.60021)].
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 951801
zbMATH Keywords: block random matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: diagonal Jordan normal form
zbMATH Keywords: random matrix spectral theory
reviewed by: A. Khorunzhy
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00356-4
OpenAlex ID: W2082977946
cites work: Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang
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cites work: Q5674306
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Wikidata QID: Q127333581
description: scientific article; zbMATH DE number 951801