Computing the determinantal representations of hyperbolic forms

Czechoslovak Mathematical Journal

Mao-Ting Chien¹  Hiroshi Nakazato² 

[1] (corresponding author), Department of Mathematics, Soochow University, 70 Linshi Road, Taipei 11102, Taiwan,;, Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, 1-bunkyocho Hirosaki-shi Aomori-ken 036-8561, Japan,

关键词: determinantal representation;    hyperbolic form;    Riemann theta function;    numerical range;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

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.

【 授权许可】

Unknown