Numerical ranges and Geršgorin discs

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

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish
Pages (from-to)1170-1192
Number of pages23
JournalLinear Algebra and Its Applications
Issue number3
StatePublished - 1 Feb 2013


  • Geršgorin disc
  • Numerical range
  • Permutationally irreducible matrix
  • Unitarily irreducible matrix


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