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

    Fingerprint

    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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