Crawford numbers of companion matrices

Hwa Long Gau, Kuo-Zhong Wang, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The (generalized) Crawford number C(A) of an n-by-n complex matrix A is, by definition, the distance from the origin to the boundary of the numerical range W(A) of A. If A is a companion matrix (Formula Presented) then it is easily seen that C(A) ≥ cos(π/n). The main purpose of this paper is to determine when the equality C(A) = cos(π/n) holds. A sufficient condition for this is that the boundary of W(A) contains a point λ for which the subspace of ℂn spanned by the vectors x with 〈Ax,x〉 = λ||x||2 has dimension 2, while a necessary condition is (Formula Presented) for some real θ. Examples are given showing that in general these conditions are not simultaneously necessary and sufficient. We then prove that they are if A is (unitarily) reducible. We also establish a lower bound for the numerical radius w(A) of A: w(A) ≥ cos(π/(n+1)), and show that the equality holds if and only if A is equal to the n-by-n Jordan block.

Original languageEnglish
Article numberoam-10-49
Pages (from-to)863-880
Number of pages18
JournalOperators and Matrices
Volume10
Issue number4
DOIs
StatePublished - 1 Dec 2016

Keywords

  • Companion matrix
  • Crawford number
  • Numerical range

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