Inverse numerical range and Abel-Jacobi map of Hermitian determinantal representation

Let A be an n x n matrix. The Hermitian parts of A are denoted by R(A) = (A + A*)/2and J(A) = (A - A*)/(2i). The kernel vectors of the linear pencil xR(A) + yJ(A) + zI(n) play a role for the inverse numerical range of A. This kernel vector technique was applied to perform the inverse numerical range of 3 x3 symmetric matrices. In this paper, we follow the kernel vector method and apply the Abel theorem for 3 x3 Hermitian matrices. We present the elliptic curve group structure of the cubic curve associated to the ternary form of the matrix, and characterize the Abel type additive structure of the divisors of the cubic curve. A numerical example is given to illustrate the characterization related to the Riemann theta representation. (C) 2021 Elsevier Inc. All rights reserved.