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

TL;DR

This paper investigates the continuity properties of the inverse of the field of values generating function for matrices, identifying conditions under which strong and weak continuity hold or fail across different matrix sizes and types.

Contribution

It characterizes the strong and weak continuity of the inverse field of values function for matrices, providing new examples of where these properties fail.

Findings

Strong continuity holds on the interior and certain boundary points of the field of values.

Counterexamples show failure of strong continuity at some boundary points for specific matrices.

Weak continuity can fail for larger matrices, with explicit examples provided.

Abstract

For an $n$-by-$n$ matrix $A$, let $f_A$ be its "field of values generating function" defined as $f_A\colon x\mapsto x^*Ax$. We consider two natural versions of the continuity, which we call strong and weak, of $f_A^{-1}$ (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 $z \in \partial F(A)$ which are either corner points, belong to the relative interior of flat portions of $\partial F(A)$, or whose preimage under $f_A$ is contained in a one-dimensional set. Consequently, $f_A^{-1}$ 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 $f_A^{-1}$ fails at certain points of $\partial F(A)$ 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.