Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials

Abstract

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.W (T ) = {ξ * T ξ : ξ ∈ C n , ξ * ξ = 1}.

Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †

“…Plaumann et al [8] mentioned the number of unitarily inequivalent classes of real symmetric matrices (S 1 , S 2 ) satisfying F S 1 +iS 2 (x, y, z) = F(x, y, z) is 2 g if the curve F(x, y, z) = 0 has no singular points, where g is the genus of the curve. For certain irreducible curve F(x, y, z) = 0 of degree 4 having singular points with genus 1, it is shown in [9], see also [10], that there are infinitely many inequivalent classes of S satisfying F S (x, y, z) = F(x, y, z). Typical hyperbolic ternary forms may admit determinantal representation by special matrices.…”