×

zbMATH — the first resource for mathematics

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.

MSC:
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
Software:
Maple; Mathematica
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] M. T. Chien, H. Nakazato: Elliptic modular invariants and numerical ranges. Linear Multilinear Algebra 63 (2015), 1501–1519. · Zbl 1314.14056 · doi:10.1080/03081087.2014.947982
[2] M. T. Chien, H. Nakazato: Determinantal representation of trigonometric polynomial curves via Sylvester method. Banach J. Math. Anal. 8 (2014), 269–278. · Zbl 1283.15025 · doi:10.15352/bjma/1381782099
[3] M. T. Chien, H. Nakazato: Singular points of cyclic weighted shift matrices. Linear Algebra Appl. 439 (2013), 4090–4100. · Zbl 1283.15117 · doi:10.1016/j.laa.2013.10.012
[4] M. T. Chien, H. Nakazato: Reduction of the c-numerical range to the classical numerical range. Linear Algebra Appl. 434 (2011), 615–624. · Zbl 1210.15023 · doi:10.1016/j.laa.2010.09.010
[5] B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, M. Schmies: Computing Riemann theta functions. Math. Comput. 73 (2004), 1417–1442. · Zbl 1092.33018 · doi:10.1090/S0025-5718-03-01609-0
[6] B. Deconinck, M. van Hoeji: Computing Riemann matrices of algebraic curves. Physica D 152–153 (2001), 28–46. · Zbl 1054.14079 · doi:10.1016/S0167-2789(01)00156-7
[7] M. Fiedler: Pencils of real symmetric matrices and real algebraic curves. Linear Algebra Appl. 141 (1990), 53–60. · Zbl 0709.15009 · doi:10.1016/0024-3795(90)90308-Y
[8] M. Fiedler: Geometry of the numerical range of matrices. Linear Algebra Appl. 37 (1981), 81–96. · Zbl 0452.15024 · doi:10.1016/0024-3795(81)90169-5
[9] J. W. Helton, I. M. Spitkovsky: The possible shapes of numerical ranges. Oper. Matrices 6 (2012), 607–611. · Zbl 1270.15014 · doi:10.7153/oam-06-41
[10] J. W. Helton, V. Vinnikov: Linear matrix inequality representations of sets. Commun. Pure Appl. Math. 60 (2007), 654–674. · Zbl 1116.15016 · doi:10.1002/cpa.20155
[11] A. Hurwitz, R. Courant: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Grundlehren der mathematischenWissenschaften. Band 3, Springer, Berlin, 1964. (In German.)
[12] R. Kippenhahn: Über den Wertevorrat einer Matrix. Math. Nachr. 6 (1951), 193–228. (In German.) · Zbl 0044.16201 · doi:10.1002/mana.19510060306
[13] P. D. Lax: Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 11 (1958), 175–194. · Zbl 0086.01603 · doi:10.1002/cpa.3160110203
[14] M. Namba: Geometry of Projective Algebraic Curves. Pure and AppliedMathematics 88, Marcel Dekker, New York, 1984. · Zbl 0556.14012
[15] D. Plaumann, B. Sturmfels, C. Vinzant: Computing linear matrix representations of HeltonVinnikov curves. Mathematical Methods in Systems, Optimization, and Control (H. Dym et al., ed.). Festschrift in honor of J. William Helton. Operator Theory: Advances and Applications 222, Birkhäuser, Basel, 2012, pp. 259–277.
[16] R. J. Walker: Algebraic Curves. Princeton Mathematical Series 13, Princeton University Press, Princeton, 1950. · Zbl 0039.37701
[17] Z. X. Wang, D. R. Guo: Special Functions. World Scientific Publishing, Teaneck, 1989.
[18] S. Wolfram: The Mathematica Book. Wolfram Media, Cambridge University Press, Cambridge, 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.