Czechoslovak Mathematical Journal

, Volume 66, Issue 3, pp 633–651 | Cite as

Computing the determinantal representations of hyperbolic forms

  • Mao-Ting ChienEmail author
  • Hiroshi Nakazato


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.


determinantal representation hyperbolic form Riemann theta function numerical range 

MSC 2010

14Q05 15A60 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  1. 1.Department of MathematicsSoochow UniversityTaipeiTaiwan
  2. 2.Department of Mathematical Sciences, Faculty of Science and TechnologyHirosaki UniversityAomori-kenJapan

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