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Computing the determinantal representations of hyperbolic forms. (English) Zbl 1413.14005
Summary: 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.

14Q05 Computational aspects of algebraic curves
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
14H42 Theta functions and curves; Schottky problem
Maple; Mathematica
Full Text: DOI Link
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