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Keywords:
determinantal representation; hyperbolic form; Riemann theta function; numerical range
determinantal representation; hyperbolic form; Riemann theta function; numerical range
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.
References:
[1] Chien, M. T., Nakazato, H.: Elliptic modular invariants and numerical ranges. Linear Multilinear Algebra 63 (2015), 1501-1519. DOI 10.1080/03081087.2014.947982 | MR 3304989 | Zbl 1314.14056
[2] Chien, M. T., Nakazato, H.: Determinantal representation of trigonometric polynomial curves via Sylvester method. Banach J. Math. Anal. 8 (2014), 269-278. DOI 10.15352/bjma/1381782099 | MR 3161694 | Zbl 1283.15025
[3] Chien, M. T., Nakazato, H.: Singular points of cyclic weighted shift matrices. Linear Algebra Appl. 439 (2013), 4090-4100. MR 3133478 | Zbl 1283.15117
[4] Chien, M. T., Nakazato, H.: Reduction of the $c$-numerical range to the classical numerical range. Linear Algebra Appl. 434 (2011), 615-624. MR 2746068 | Zbl 1210.15023
[5] Deconinck, B., Heil, M., Bobenko, A., Hoeij, M. van, Schmies, M.: Computing Riemann theta functions. Math. Comput. 73 (2004), 1417-1442. DOI 10.1090/S0025-5718-03-01609-0 | MR 2047094
[6] Deconinck, B., Hoeji, M. van: Computing Riemann matrices of algebraic curves. Physica D 152-153 (2001), 28-46. MR 1837895
[7] Fiedler, M.: Pencils of real symmetric matrices and real algebraic curves. Linear Algebra Appl. 141 (1990), 53-60. MR 1076103 | Zbl 0709.15009
[8] Fiedler, M.: Geometry of the numerical range of matrices. Linear Algebra Appl. 37 (1981), 81-96. DOI 10.1016/0024-3795(81)90169-5 | MR 0636211 | Zbl 0452.15024
[9] Helton, J. W., Spitkovsky, I. M.: The possible shapes of numerical ranges. Oper. Matrices 6 (2012), 607-611. DOI 10.7153/oam-06-41 | MR 2987030 | Zbl 1270.15014
[10] Helton, J. W., Vinnikov, V.: Linear matrix inequality representations of sets. Commun. Pure Appl. Math. 60 (2007), 654-674. DOI 10.1002/cpa.20155 | MR 2292953
[11] Hurwitz, A., Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Grundlehren der mathematischen Wissenschaften. Band 3 Springer, Berlin German (1964). MR 0173749 | Zbl 0135.12101
[12] Kippenhahn, R.: Über den Wertevorrat einer Matrix. German Math. Nachr. 6 (1951), 193-228. DOI 10.1002/mana.19510060306 | MR 0059242 | Zbl 0044.16201
[13] Lax, P. D.: Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 11 (1958), 175-194. MR 0098110 | Zbl 0086.01603
[14] Namba, M.: Geometry of Projective Algebraic Curves. Pure and Applied Mathematics 88 Marcel Dekker, New York (1984). MR 0768929 | Zbl 0556.14012
[15] Plaumann, D., Sturmfels, B., Vinzant, C.: Computing linear matrix representations of Helton-Vinnikov curves. H. Dym et al. Mathematical Methods in Systems, Optimization, and Control Festschrift in honor of J. William Helton. Operator Theory: Advances and Applications 222 Birkhäuser, Basel 259-277 (2012). MR 2962788 | Zbl 1328.14093
[16] Walker, R. J.: Algebraic Curves. Princeton Mathematical Series 13 Princeton University Press, Princeton (1950). MR 0033083 | Zbl 0039.37701
[17] Wang, Z. X., Guo, D. R.: Special Functions. World Scientific Publishing, Teaneck (1989). MR 1034956 | Zbl 0724.33001
[18] Wolfram, S.: The Mathematica Book. Wolfram Media, Cambridge University Press, Cambridge (1996). MR 1404696 | Zbl 0878.65001
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