Numerical ranges and Geršgorin discs

Chi Tung Chang, Hwa Long Gau, Kuo Zhong Wang, Pei Yuan Wu

研究成果: 雜誌貢獻期刊論文同行評審


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).

頁(從 - 到)1170-1192
期刊Linear Algebra and Its Applications
出版狀態已出版 - 1 2月 2013


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