### Abstract

For two n-by-n matrices, A,B, the product field of values is the set P(A,B)={〈Ax,x〉〈Bx,x〉:x∈^{Cn},||x||=1}. In this paper, we establish basic properties of the product field of values. The main results are a proof that the product field is a simply connected subset of the complex plane and a characterization of matrix pairs for which the product field has nonempty interior.

Original language | English (US) |
---|---|

Pages (from-to) | 2155-2173 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 438 |

Issue number | 5 |

DOIs | |

State | Published - Mar 1 2013 |

### Fingerprint

### Keywords

- Field of values
- Numerical range

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*438*(5), 2155-2173. https://doi.org/10.1016/j.laa.2012.09.028

**The product field of values.** / Corey, Daniel; Johnson, Charles R.; Kirk, Ryan; Lins, Brian; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 438, no. 5, pp. 2155-2173. https://doi.org/10.1016/j.laa.2012.09.028

}

TY - JOUR

T1 - The product field of values

AU - Corey, Daniel

AU - Johnson, Charles R.

AU - Kirk, Ryan

AU - Lins, Brian

AU - Spitkovsky, Ilya

PY - 2013/3/1

Y1 - 2013/3/1

N2 - For two n-by-n matrices, A,B, the product field of values is the set P(A,B)={〈Ax,x〉〈Bx,x〉:x∈Cn,||x||=1}. In this paper, we establish basic properties of the product field of values. The main results are a proof that the product field is a simply connected subset of the complex plane and a characterization of matrix pairs for which the product field has nonempty interior.

AB - For two n-by-n matrices, A,B, the product field of values is the set P(A,B)={〈Ax,x〉〈Bx,x〉:x∈Cn,||x||=1}. In this paper, we establish basic properties of the product field of values. The main results are a proof that the product field is a simply connected subset of the complex plane and a characterization of matrix pairs for which the product field has nonempty interior.

KW - Field of values

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=84872100952&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872100952&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.09.028

DO - 10.1016/j.laa.2012.09.028

M3 - Article

VL - 438

SP - 2155

EP - 2173

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 5

ER -