### Abstract

For an n-by-n matrix A, let fA be its 'field of values generating function' defined as f_{A}: x {mapping} x*Ax. We consider two natural versions of the continuity, which we call strong and weak, of f_{A}
^{-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 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 ∂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 language | English (US) |
---|---|

Pages (from-to) | 1329-1338 |

Number of pages | 10 |

Journal | Linear and Multilinear Algebra |

Volume | 61 |

Issue number | 10 |

DOIs | |

State | Published - Dec 16 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Linear and Multilinear Algebra*,

*61*(10), 1329-1338. https://doi.org/10.1080/03081087.2012.736991

**Continuity properties of vectors realizing points in the classical field of values.** / Corey, Dan; Johnson, Charles R.; Kirk, Ryan; Lins, Brian; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear and Multilinear Algebra*, vol. 61, no. 10, pp. 1329-1338. https://doi.org/10.1080/03081087.2012.736991

}

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

PY - 2013/12/16

Y1 - 2013/12/16

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

UR - http://www.scopus.com/inward/record.url?scp=84890061753&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890061753&partnerID=8YFLogxK

U2 - 10.1080/03081087.2012.736991

DO - 10.1080/03081087.2012.736991

M3 - Article

AN - SCOPUS:84890061753

VL - 61

SP - 1329

EP - 1338

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 10

ER -