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- W4311014087 abstract "For an $n$-by-$n$ matrix $A$, let $f_A$ be its field of values generating function defined as $f_Acolon xmapsto x^*Ax$. 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." @default.
- W4311014087 created "2022-12-22" @default.
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- W4311014087 date "2013-07-18" @default.
- W4311014087 modified "2023-10-16" @default.
- W4311014087 title "Continuity properties of vectors realizing points in the classical field of values" @default.
- W4311014087 doi "https://doi.org/10.48550/arxiv.1307.5033" @default.
- W4311014087 hasPublicationYear "2013" @default.
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