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 ℜ ( A ) + y ℑ ( A ) + z I 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.

Keywords

• determinantal representation
• hyperbolic ternary forms
• rational curves
• toeplitz matrices
• numerical range