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Symmetry 2018, 10(3), 55; https://doi.org/10.3390/sym10030055

Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices

1
Department of Mathematics, Soochow University, Taipei 11102, Taiwan
2
Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan
Portions of this article were presented at The First International Conference on Symmetry, Barcelona, 16–18 October 2017.
Portions of this article were presented at The First International Conference on Symmetry, Barcelona, 16–18 October 2017.
*
Author to whom correspondence should be addressed.
Received: 5 January 2018 / Revised: 20 February 2018 / Accepted: 22 February 2018 / Published: 27 February 2018
Full-Text   |   PDF [273 KB, uploaded 28 February 2018]

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 ) + 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. View Full-Text
Keywords: determinantal representation; hyperbolic ternary forms; rational curves; toeplitz matrices; numerical range determinantal representation; hyperbolic ternary forms; rational curves; toeplitz matrices; numerical range
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Chien, M.-T.; Nakazato, H. Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices . Symmetry 2018, 10, 55.

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