Crawford numbers of powers of a matrix

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

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

or an n-by-n matrix A, its Crawford number c(A) (resp., generalized Crawford number C(A)) is, by definition, the distance from the origin to its numerical range W(A) (resp., the boundary of its numerical range ∂W(A)). It is shown that if A has eigenvalues λ1,⋯, λn arranged so that |λ 1|≥⋯≥|λn|, then lim kc(Ak)1/k (resp., limkC(A k)1/k) equals 0 or |λn| (resp., |λj| for some j, 1≤j≤n). For a normal A, more can be said, namely, lim kc(A(Ak)1/k=|λn| (resp., limkC((Ak)1/k=|λj| for some j, 3≤j≤n). In these cases, the above possible values can all be assumed by some A.

Original languageEnglish
Pages (from-to)2243-2254
Number of pages12
JournalLinear Algebra and Its Applications
Volume433
Issue number11-12
DOIs
StatePublished - 30 Dec 2010

Keywords

  • Crawford number
  • Generalized Crawford number
  • Numerical range

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