Higher-rank numerical ranges and Kippenhahn polynomials

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

We prove that two n-by-n matrices A and B have their rank-k numerical ranges Λk(A) and Λk(B) equal to each other for all k,1≤k≤⌊n/2⌋+1, if and only if their Kippenhahn polynomials pA(x,y,z)≡det(xReA+yImA+zIn) and pB(x,y,z)≡det(xReB+yImB+zIn) coincide. The main tools for the proof are the Li-Sze characterization of higher-rank numerical ranges, Weyl's perturbation theorem for eigenvalues of Hermitian matrices and Bézout's theorem for the number of common zeros for two homogeneous polynomials.

Original languageEnglish
Pages (from-to)3054-3061
Number of pages8
JournalLinear Algebra and Its Applications
Volume438
Issue number7
DOIs
StatePublished - 1 Apr 2013

Keywords

  • Higher-rank numerical range
  • Kippenhahn polynomial

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