TY - JOUR

T1 - Continuity properties of vectors realizing points in the classical field of values

AU - Corey, Dan

AU - Johnson, Charles R.

AU - Kirk, Ryan

AU - Lins, Brian

AU - Spitkovsky, Ilya

N1 - Funding Information:
This work was partially supported by NSF grant DMS-0751964.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Field of values

KW - Inverse continuity

KW - Numerical range

KW - Weak continuity

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U2 - 10.1080/03081087.2012.736991

DO - 10.1080/03081087.2012.736991

M3 - Article

AN - SCOPUS:84890061753

SN - 0308-1087

VL - 61

SP - 1329

EP - 1338

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 10

ER -