Annals of Functional Analysis

Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials

Mao-Ting Chien and Hiroshi Nakazato

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The numerical range of a matrix, according to Kippenhahn, is determined by a hyperbolic determinantal form of linear Hermitian matrices associated to the matrix. On the other hand, using Riemann theta functions, Helton and Vinnikov confirmed that a hyperbolic form always admits a determinantal representation of linear real symmetric matrices. The Riemann matrix of the hyperbolic curve plays the main role in the existence of real symmetric matrices. In this article, we implement computations of the Riemann matrix and the Abel–Jacobi variety of the hyperbolic curve associated to a determinantal polynomial of a matrix. Further, we prove that the lattice of the Abel–Jacobi variety is decomposed into the direct sum of two orthogonal lattices for some 4×4 Toeplitz matrices.

Article information

Ann. Funct. Anal., Volume 8, Number 2 (2017), 152-167.

Received: 25 May 2016
Accepted: 12 August 2016
First available in Project Euclid: 14 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
Secondary: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 14H55: Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]

Riemann matrix period matrix lattice numerical range


Chien, Mao-Ting; Nakazato, Hiroshi. Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials. Ann. Funct. Anal. 8 (2017), no. 2, 152--167. doi:10.1215/20088752-3773229.

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