Continuity properties of vectors realizing points in the classical field of values
Dan Corey, Charles R. Johnson, Ryan Kirk, Brian Lins, Ilya M. Spitkovsky · Linear and Multilinear Algebra · 2012
For an n-by-n matrix A, let f A be its ‘field of values generating function’ defined as f A : x ↦ x*Ax. We consider two natural versions of the continuity, which we call strong and weak, of (which is of course multi-valued) 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 f A is contained in a one-dimensional set. Consequently, 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 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.