Title: Numerical ranges of weighted shift matrices with periodic weights
Authors: Tsai, Ming Cheng
Department of Applied Mathematics
Keywords: Numerical range;Weighted shift matrix;Periodic weights;Degree-n homogeneous polynomial;Reducible matrix
Issue Date: 1-Nov-2011
Abstract: Let A be an n-by-n (n >= 2) matrix of the form [0 a(1) 0 a(n-1) a(n) 0] We show that if the a(j)'s are nonzero and their moduli are periodic, then the boundary of its numerical range contains a line segment. We also prove that partial derivative W (A) contains a noncircular elliptic arc if and only if the a(j)'s are nonzero, n is even, vertical bar a(1)vertical bar = vertical bar a(3)vertical bar = ... = vertical bar a(n-1)vertical bar, vertical bar a(2)vertical bar = vertical bar a(4)vertical bar = ... = vertical bar a(n)vertical bar and vertical bar a(1)vertical bar not equal vertical bar a(2)vertical bar. Finally, we give a criterion for A to be reducible and completely characterize the numerical ranges of such matrices. (C) 2011 Elsevier Inc. All rights reserved.
URI: http://dx.doi.org/10.1016/j.laa.2011.04.028
ISSN: 0024-3795
DOI: 10.1016/j.laa.2011.04.028
Volume: 435
Issue: 9
Begin Page: 2296
End Page: 2302
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