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Bibliographic description

punktacja MNiSW [2014 A]: 35 

title:	Numerical range for random matrices
author:	Collins Benoît, Gawron Piotr, Litvak Alexander E., Życzkowski Karol
journal title:	Journal of Mathematical Analysis and Applications
volume:	418
issue:	1
date of publication :	2014
pages:	516-533
ISSN:	0022-247X
eISSN:	1096-0813
DOI:	10.1016/j.jmaa.2014.03.072
language:	English
journal language:	English
abstract in English:	We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius √2. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width √2 - 1 containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius √2, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to √2e.
keywords in English:	Ginibre ensemble, Gaussian random matrix, GUE, numerical range, field of values, triangular random matrix
number of pulisher's sheets:	1
affiliation:	Wydział Fizyki, Astronomii i Informatyki Stosowanej : Instytut Fizyki im. Mariana Smoluchowskiego
type:	journal article
subtype:	academic paper

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