TY - JOUR

T1 - Numerical ranges and Geršgorin discs

AU - Chang, Chi Tung

AU - Gau, Hwa Long

AU - Wang, Kuo Zhong

AU - Wu, Pei Yuan

N1 - Funding Information:
Corresponding author. E-mail addresses: [email protected] (C.-T. Chang), [email protected] (H.-L. Gau), [email protected] (K.-Z. Wang), [email protected] (P.Y. Wu). 1 Research supported by the National Science Council of the Republic of China under the post-doctoral fellowship NSC 100-2811-M-009-054. 2 Research supported by the National Science Council of the Republic of China under the projects NSC 100-2115-M-008-004, NSC 99-2115-M-009-013-MY2 and NSC 99-2115-M-009-002-MY2, respectively.

PY - 2013/2/1

Y1 - 2013/2/1

N2 - For a complex matrix A=[aij]i,j=1n, let W(A) be its numerical range, and let G(A) be the convex hull of x i=1n{z∈C:|z- aij|≤(∑ i≠j(| aij|+| aij|))/2} and G'(A)=x{G(U *AU):Un-by-nunitary}.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).

AB - For a complex matrix A=[aij]i,j=1n, let W(A) be its numerical range, and let G(A) be the convex hull of x i=1n{z∈C:|z- aij|≤(∑ i≠j(| aij|+| aij|))/2} and G'(A)=x{G(U *AU):Un-by-nunitary}.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).

KW - Geršgorin disc

KW - Numerical range

KW - Permutationally irreducible matrix

KW - Unitarily irreducible matrix

UR - http://www.scopus.com/inward/record.url?scp=84870387622&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.09.003

DO - 10.1016/j.laa.2012.09.003

M3 - 期刊論文

AN - SCOPUS:84870387622

SN - 0024-3795

VL - 438

SP - 1170

EP - 1192

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 3

ER -