Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Chi Kwong Li, Ilya Spitkovsky, Sudheer Shukla

Research output: Contribution to journalArticlepeer-review

Abstract

Let Mn be the algebra of all n × n complex matrices. For 1 ≤ k ≤ n, the kth numerical range of A ∈ Mn is defined by Wk(A) = {(1/k)Σjk=1 xj*Axj : {x1, . . . , xk} is an orthonormal set in ℂn}. It is known that {tr A/n} = Wn(A) ⊆ Wn-1(A) ⊆ ⋯ ⊆ W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 ≤ m < k ≤ n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.

Original languageEnglish (US)
Pages (from-to)323-349
Number of pages27
JournalLinear Algebra and Its Applications
Volume270
Issue number1-3
DOIs
StatePublished - Feb 1998

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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