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 journalArticlepeer-review


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
Issue number10
StatePublished - 2013


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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