Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †

Abstract

Let A be an n × n complex matrix. Assume the determinantal curve V A = {[(x, y, z)] ∈ CP 2 : F A (x, y, z) = det(x (A) + y (A) + zI n) = 0} is a rational curve. The Fiedler formula provides a complex symmetric matrix S satisfying F S (x, y, z) = F A (x, y, z). It is also known that every Toeplitz matrix is unitarily similar to a symmetric matrix. In this paper, we investigate the unitary similarity of the symmetric matrix S and the matrix A in the Fiedler theorem for a specific parametrized family of 4 × 4 nilpotent Toeplitz matrices A. We show that there are either one or at least three unitarily inequivalent symmetric matrices which admit the determinantal representation of the ternary from F A (x, y, z) associated to the specific 4 × 4 nilpotent Toeplitz matrices.