### Abstract

The ratio field of values, a generalization of the classical field of values to a pair of n-by-n matrices, is defined and studied, primarily from a geometric point of view. Basic functional properties of the ratio field are developed and used. A decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation. Primary theoretical results include the following. It is shown (1) for which denominator matrices the ratio field is always convex, (2) certain other cases of convex pairs are given, and (3) that, at least for n=2, the ratio field obeys a near convexity property that the intersection with any line segment has at most n components. Generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated. It is shown that generally such extensions need not be convex or even simply connected.

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

Pages (from-to) | 1119-1136 |

Number of pages | 18 |

Journal | Linear Algebra and Its Applications |

Volume | 434 |

Issue number | 4 |

DOIs | |

State | Published - Feb 15 2011 |

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### Keywords

- Field of values
- Generalized eigenvalue problem
- Matrix pencil
- Matrix polynomial
- 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*,

*434*(4), 1119-1136. https://doi.org/10.1016/j.laa.2010.10.022

**The ratio field of values.** / Einstein, Eduard; Johnson, Charles R.; Lins, Brian; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 434, no. 4, pp. 1119-1136. https://doi.org/10.1016/j.laa.2010.10.022

}

TY - JOUR

T1 - The ratio field of values

AU - Einstein, Eduard

AU - Johnson, Charles R.

AU - Lins, Brian

AU - Spitkovsky, Ilya

PY - 2011/2/15

Y1 - 2011/2/15

N2 - The ratio field of values, a generalization of the classical field of values to a pair of n-by-n matrices, is defined and studied, primarily from a geometric point of view. Basic functional properties of the ratio field are developed and used. A decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation. Primary theoretical results include the following. It is shown (1) for which denominator matrices the ratio field is always convex, (2) certain other cases of convex pairs are given, and (3) that, at least for n=2, the ratio field obeys a near convexity property that the intersection with any line segment has at most n components. Generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated. It is shown that generally such extensions need not be convex or even simply connected.

AB - The ratio field of values, a generalization of the classical field of values to a pair of n-by-n matrices, is defined and studied, primarily from a geometric point of view. Basic functional properties of the ratio field are developed and used. A decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation. Primary theoretical results include the following. It is shown (1) for which denominator matrices the ratio field is always convex, (2) certain other cases of convex pairs are given, and (3) that, at least for n=2, the ratio field obeys a near convexity property that the intersection with any line segment has at most n components. Generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated. It is shown that generally such extensions need not be convex or even simply connected.

KW - Field of values

KW - Generalized eigenvalue problem

KW - Matrix pencil

KW - Matrix polynomial

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=78650510612&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650510612&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.10.022

DO - 10.1016/j.laa.2010.10.022

M3 - Article

AN - SCOPUS:78650510612

VL - 434

SP - 1119

EP - 1136

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 4

ER -