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 |