Computing the determinantal representations of hyperbolic forms

Mao-Ting Chien; Hiroshi Nakazato

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 633-651
  • ISSN: 0011-4642

Abstract

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The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1 . We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0 , 1 , and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.

How to cite

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Chien, Mao-Ting, and Nakazato, Hiroshi. "Computing the determinantal representations of hyperbolic forms." Czechoslovak Mathematical Journal 66.3 (2016): 633-651. <http://eudml.org/doc/286803>.

@article{Chien2016,
abstract = {The numerical range of an $n\times n$ matrix is determined by an $n$ degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an $n$ degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus $g=1$. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus $g=0,1$, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.},
author = {Chien, Mao-Ting, Nakazato, Hiroshi},
journal = {Czechoslovak Mathematical Journal},
keywords = {determinantal representation; hyperbolic form; Riemann theta function; numerical range},
language = {eng},
number = {3},
pages = {633-651},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Computing the determinantal representations of hyperbolic forms},
url = {http://eudml.org/doc/286803},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Chien, Mao-Ting
AU - Nakazato, Hiroshi
TI - Computing the determinantal representations of hyperbolic forms
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 633
EP - 651
AB - The numerical range of an $n\times n$ matrix is determined by an $n$ degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an $n$ degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus $g=1$. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus $g=0,1$, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.
LA - eng
KW - determinantal representation; hyperbolic form; Riemann theta function; numerical range
UR - http://eudml.org/doc/286803
ER -

References

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