Spherical ensembles
type: Item
instance of: scholarly article
title: Spherical ensembles
zbMATH Open document ID: 1269.15037
author: José A. Díaz-García
author: Ramón Gutiérrez-Jáimez
published in: Linear Algebra and its Applications
publication date: 2013-04-12T00:00:00Z
review text: The authors give a unifying study regarding densities of random matrices with entries being real (\(\beta=1\)), complex (\(\beta=2\)), quaternion (\(\beta=4\)) or octonion (\(\beta=8\)) numbers. The kind of matrices studied in the paper are characterized by the so-called spherical and elliptical distributions, the first ones are defined through an invariant property when multiplying, for example in the complex case, by unitary matrices; and the second ones are defined through a special form of the distribution on the space of matrices. One instance of this kind of ensembles is when imposing independence of the entries, in which case the entries should be normal random variables. The first results describe the distributions of the ensembles and the so-called joint element densities. The authors then turn into the study of certain constructions of the form \((M+M^{*})/2\) or \(M^{*}M\), where \(M^{*}\) represets the conjugate transpose matrix; this is related to what is known as the Hermite, Laguerre, Jacobi and Fourier ensembles, and the authors use the term vector-spherical ensembles in their definition. In turn, they are able to describe eigenvalue densities of these random matrices. In the last part of the paper, the authors also describe densities associated to some symmetric tridiagonal random matrices related to their vector-spherical ensembles, with the parameter \(\beta\) belonging to \([0,\infty)\).
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 60B20
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 6153403
zbMATH Keywords: real random matrices
zbMATH Keywords: complex random matrices
zbMATH Keywords: quaternion random matrices
zbMATH Keywords: octonion random matrices
zbMATH Keywords: beta-ensemble
zbMATH Keywords: Jacobians
zbMATH Keywords: spherical-ensemble
zbMATH Keywords: Hermite-ensemble
zbMATH Keywords: Laguerre-ensemble
zbMATH Keywords: Jacobi-ensemble
zbMATH Keywords: eigenvalue densities
reviewed by: Carlos Gabriel Pacheco
MaRDI profile type: Publication
full work available at URL: j.laa.2012.12.038
OpenAlex ID: W4205474755
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DOI: 10.1016/J.LAA.2012.12.038
description: scientific article; zbMATH DE number 6153403