Continuity properties of vectors realizing points in the classical field of values

Dan Corey, Charles R. Johnson, Ryan Kirk, Brian Lins, Ilya Spitkovsky

Research output: Contribution to journalArticle

Abstract

For an n-by-n matrix A, let fA be its 'field of values generating function' defined as fA: x {mapping} x*Ax. We consider two natural versions of the continuity, which we call strong and weak, of fA -1 (which is of course multivalued) on the field of values F(A). The strong continuity holds, in particular, on the interior of F(A), and at such points z ∈ ∂F(A) which are either corner points, belong to the relative interior of flat portions of ∂F(A), or whose preimage under fA is contained in a one-dimensional set. Consequently, fA -1 is continuous in this sense on the whole F(A) for all normal, 2-by-2, and unitarily irreducible 3-by-3 matrices. Nevertheless, we show by example that the strong continuity of fA -1 fails at certain points of ∂F(A) for some (unitarily reducible) 3-by-3 and (unitarily irreducible) 4-by-4 matrices. The weak continuity, in its turn, fails for some unitarily reducible 4-by-4 and untiarily irreducible 6-by-6 matrices.

Original languageEnglish (US)
Pages (from-to)1329-1338
Number of pages10
JournalLinear and Multilinear Algebra
Volume61
Issue number10
DOIs
StatePublished - Dec 16 2013

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Field of Values
Interior
Weak Continuity
Multivalued
Value Function
Generating Function

Keywords

  • Field of values
  • Inverse continuity
  • Numerical range
  • Weak continuity

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Continuity properties of vectors realizing points in the classical field of values. / Corey, Dan; Johnson, Charles R.; Kirk, Ryan; Lins, Brian; Spitkovsky, Ilya.

In: Linear and Multilinear Algebra, Vol. 61, No. 10, 16.12.2013, p. 1329-1338.

Research output: Contribution to journalArticle

Corey, Dan ; Johnson, Charles R. ; Kirk, Ryan ; Lins, Brian ; Spitkovsky, Ilya. / Continuity properties of vectors realizing points in the classical field of values. In: Linear and Multilinear Algebra. 2013 ; Vol. 61, No. 10. pp. 1329-1338.
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