Companion matrices: Reducibility, numerical ranges and similarity to contractions

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


In this paper, we study some unitary-equivalence properties of the companion matrices. We obtain a criterion for a companion matrix to be reducible and show that the numerical range of a companion matrix is a circular disc centered at the origin if and only if the matrix equals the (nilpotent) Jordan block. However, the more general assertion that a companion matrix is determined by its numerical range turns out to be false. We also determine, for an n×n matrix A with eigenvalues in the open unit disc, the defect index of a contraction to which A is similar.

Original languageEnglish
Pages (from-to)127-142
Number of pages16
JournalLinear Algebra and Its Applications
Issue number1-3 SPEC. ISS.
StatePublished - 15 May 2004


  • Companion matrix
  • Defect index
  • Numerical range
  • Reducible matrix


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