When is the numerical range of a nilpotent matrix circular?
type: Item
instance of: scholarly article
title: When is the numerical range of a nilpotent matrix circular?
zbMATH Open document ID: 1191.15018
author: V. Matache
author: Mihaela T. Matache
published in: Applied Mathematics and Computation
publication date: 2010-03-31T00:00:00Z
review text: The problem formulated in the title is examined. A \(2\times 2\) nilpotent matrix always has circular numerical range. The situation when \(3\times 3\) nilpotent matrices have circular numerical ranges is completely described by \textit{D. S. Keeler, L. Rodman} and \textit{I. M. Spitkovsky} [Linear Algebra Appl. 252, 115--139 (1997; Zbl 0876.15020)]. The \(4\times 4\) case is covered by \textit{H.-L. Gau} [Taiwanese J. Math. 10, No.~1, 117--128 (2006; Zbl 1099.15022)] as follows: A necessary and sufficient condition that the numerical range of a nilpotent matrix \(M\) of size at most 4 be circular is that the traces \(\text{tr} M^*M^2\) and \(\text{tr} M^*M^3\) be all null. For matrices of size 5, the situation becomes more complicated. The authors show that the analog of the so-called trace condition appearing in the result by Gau does not hold in the \(5\times 5\) case, counterexamples being given. The main result of the paper completely characterizes the situation when a \(5\times 5\) nilpotent matrix has circular numerical range. Also, the case when a \(5\times 5\) nilpotent matrix satisfying, respectively non-satisfying, the trace condition has circular numerical range is discussed thoroughly.
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 5688870
zbMATH Keywords: nilpotent matrix
zbMATH Keywords: circular numerical range
zbMATH Keywords: trace condition
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: j.amc.2010.01.052
OpenAlex ID: W1963945643
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DOI: 10.1016/J.AMC.2010.01.052
description: scientific article; zbMATH DE number 5688870