Geodesic
Computing the determinantal representations of hyperbolic forms
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 633-651 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-016-0283-9
Classification : 14Q05, 15A60
Keywords: determinantal representation; hyperbolic form; Riemann theta function; numerical range 
@article{10_1007_s10587_016_0283_9,
     author = {Chien, Mao-Ting and Nakazato, Hiroshi},
     title = {Computing the determinantal representations of hyperbolic forms},
     journal = {Czechoslovak Mathematical Journal},
     pages = {633--651},
     year = {2016},
     volume = {66},
     number = {3},
     doi = {10.1007/s10587-016-0283-9},
     mrnumber = {3556858},
     zbl = {06644024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0283-9/}
}
TY  - JOUR
AU  - Chien, Mao-Ting
AU  - Nakazato, Hiroshi
TI  - Computing the determinantal representations of hyperbolic forms
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 633
EP  - 651
VL  - 66
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0283-9/
DO  - 10.1007/s10587-016-0283-9
LA  - en
ID  - 10_1007_s10587_016_0283_9
ER  - 
%0 Journal Article
%A Chien, Mao-Ting
%A Nakazato, Hiroshi
%T Computing the determinantal representations of hyperbolic forms
%J Czechoslovak Mathematical Journal
%D 2016
%P 633-651
%V 66
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0283-9/
%R 10.1007/s10587-016-0283-9
%G en
%F 10_1007_s10587_016_0283_9
Chien, Mao-Ting; Nakazato, Hiroshi. Computing the determinantal representations of hyperbolic forms. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 633-651. doi: 10.1007/s10587-016-0283-9

[1] Chien, M. T., Nakazato, H.: Elliptic modular invariants and numerical ranges. Linear Multilinear Algebra 63 (2015), 1501-1519. | DOI | MR | Zbl

[2] Chien, M. T., Nakazato, H.: Determinantal representation of trigonometric polynomial curves via Sylvester method. Banach J. Math. Anal. 8 (2014), 269-278. | DOI | MR | Zbl

[3] Chien, M. T., Nakazato, H.: Singular points of cyclic weighted shift matrices. Linear Algebra Appl. 439 (2013), 4090-4100. | MR | Zbl

[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 | Zbl

[5] Deconinck, B., Heil, M., Bobenko, A., Hoeij, M. van, Schmies, M.: Computing Riemann theta functions. Math. Comput. 73 (2004), 1417-1442. | DOI | MR

[6] Deconinck, B., Hoeji, M. van: Computing Riemann matrices of algebraic curves. Physica D 152-153 (2001), 28-46. | MR

[7] Fiedler, M.: Pencils of real symmetric matrices and real algebraic curves. Linear Algebra Appl. 141 (1990), 53-60. | MR | Zbl

[8] Fiedler, M.: Geometry of the numerical range of matrices. Linear Algebra Appl. 37 (1981), 81-96. | DOI | MR | Zbl

[9] Helton, J. W., Spitkovsky, I. M.: The possible shapes of numerical ranges. Oper. Matrices 6 (2012), 607-611. | DOI | MR | Zbl

[10] Helton, J. W., Vinnikov, V.: Linear matrix inequality representations of sets. Commun. Pure Appl. Math. 60 (2007), 654-674. | DOI | MR

[11] Hurwitz, A., Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Grundlehren der mathematischen Wissenschaften. Band 3 Springer, Berlin German (1964). | MR | Zbl

[12] Kippenhahn, R.: Über den Wertevorrat einer Matrix. German Math. Nachr. 6 (1951), 193-228. | DOI | MR | Zbl

[13] Lax, P. D.: Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 11 (1958), 175-194. | MR | Zbl

[14] Namba, M.: Geometry of Projective Algebraic Curves. Pure and Applied Mathematics 88 Marcel Dekker, New York (1984). | MR | Zbl

[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 | Zbl

[16] Walker, R. J.: Algebraic Curves. Princeton Mathematical Series 13 Princeton University Press, Princeton (1950). | MR | Zbl

[17] Wang, Z. X., Guo, D. R.: Special Functions. World Scientific Publishing, Teaneck (1989). | MR | Zbl

[18] Wolfram, S.: The Mathematica Book. Wolfram Media, Cambridge University Press, Cambridge (1996). | MR | Zbl

Cité par Sources :