標題: Numerical ranges and Gersgorin discs
作者: Chang, Chi-Tung
Gau, Hwa-Long
Wang, Kuo-Zhong
Wu, Pei Yuan
應用數學系
Department of Applied Mathematics
關鍵字: Numerical range;Gersgorin disc;Unitarily irreducible matrix;Permutationally irreducible matrix
公開日期: 1-二月-2013
摘要: For a complex matrix A = vertical bar a(ij)vertical bar(n)(i,j=1), let W(A) be its numerical range, and let G(A) be the convex hull of U-i=1(n) {z is an element of C : vertical bar z - a(ij)vertical bar <= (Sigma(i not equal j)(vertical bar a(ij)vertical bar)/2} and G'(A) = n{G(U*AU) : U n-by-n unitary). It is known that W(A) is always contained in G(A) and hence in G'(A). In this paper, we consider conditions for W (A) to be equal to G(A) or G'(A). We show that if W(A) = G'(A). then the boundary of W(A) consists only of circular arcs and line segments. If, moreover, A is unitarily irreducible, then W (A) is a circular disc. (Almost) complete characterizations of 2-by-2 and 3-by-3 matrices A for which W(A) = G'(A) are obtained. We also give criteria for the equality of W(A) and G(A). In particular, such A's among the permutationally irreducible ones must have even sizes. We also characterize those A's with size 2 or 4 which satisfy W(A) = G(A). Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
URI: http://dx.doi.org/10.1016/j.laa.2012.09.003
http://hdl.handle.net/11536/20779
ISSN: 0024-3795
DOI: 10.1016/j.laa.2012.09.003
期刊: LINEAR ALGEBRA AND ITS APPLICATIONS
Volume: 438
Issue: 3
起始頁: 1170
結束頁: 1192
顯示於類別:期刊論文


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