#
Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices ^{†}^{ †}

^{1}

^{2}

^{*}

^{†}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Symmetric Representation

**Lemma**

**1.**

- (i)
- The ternary form ${F}_{A}(x,y,z)$ is irreducible and the algebraic curve ${F}_{A}(x,y,z)=0$ is rational;
- (ii)
- The roots of ${F}_{A}(0,-1,z)=0$ are $(a+2)/2,(a-2)/2,-(a-2)/2,-(a+2)/2$;
- (iii)
- If a real symmetric matrix $C=({c}_{jk})$ satisfies ${F}_{C/2+iB/2}(x,y,z)={F}_{A}(x,y,z)$ then
- (a)
- ${c}_{11}=1,{c}_{22}=-1,{c}_{33}=-1,{c}_{44}=1$;
- (b)
- The entries ${c}_{jk}$, $1\le j<k\le 4$ are solutions of the system of equations:$$\begin{array}{ccc}\hfill {P}_{1}& =& {c}_{12}^{2}+{c}_{34}^{2}+{c}_{13}^{2}+{c}_{24}^{2}+{c}_{14}^{2}+{c}_{23}^{2}-(2{a}^{2}+6)=0,\hfill \\ \hfill {P}_{2}& =& a({c}_{12}^{2}-{c}_{34}^{2})+2({c}_{13}^{2}-{c}_{24}^{2})=0,\hfill \\ \hfill {P}_{3}& =& (4-{a}^{2})({c}_{12}^{2}+{c}_{34}^{2})+({a}^{2}-4)({c}_{13}^{2}+{c}_{24}^{2})+{(a-2)}^{2}{c}_{14}^{2}\hfill \\ & & +{(a+2)}^{2}{c}_{23}^{2}-(2{a}^{4}-14{a}^{2}+24)=0,\hfill \\ \hfill {P}_{4}& =& {c}_{12}({c}_{13}{c}_{23}+{c}_{14}{c}_{24})+{c}_{34}({c}_{13}{c}_{14}+{c}_{23}{c}_{24})+{c}_{14}^{2}-{c}_{23}^{2}-8a=0,\hfill \\ \hfill {P}_{5}& =& {c}_{12}\left(-(a+2){c}_{13}{c}_{23}+(2-a){c}_{14}{c}_{24}\right)+{c}_{34}((a-2){c}_{13}{c}_{14}\hfill \\ & & +(2+a){c}_{23}{c}_{24})-2{c}_{12}^{2}+2{c}_{34}^{2}-a{c}_{13}^{2}+a{c}_{24}^{2}=0,\hfill \\ \hfill {P}_{6}& =& {c}_{12}^{2}{c}_{34}^{2}+{c}_{13}^{2}{c}_{24}^{2}+{c}_{14}^{2}{c}_{23}^{2}-2{c}_{12}{c}_{34}({c}_{14}{c}_{23}+{c}_{13}{c}_{24})\hfill \\ & & -2{c}_{13}{c}_{24}{c}_{14}{c}_{23}+2{c}_{12}({c}_{13}{c}_{23}-{c}_{14}{c}_{24})+2{c}_{34}({c}_{23}{c}_{24}-{c}_{13}{c}_{14})\hfill \\ & & +{c}_{12}^{2}+{c}_{34}^{2}-{c}_{14}^{2}-{c}_{23}^{2}+{c}_{13}^{2}+{c}_{24}^{2}-{a}^{4}+8{a}^{2}+1=0,\hfill \end{array}$$

and $B=\mathrm{diag}(a+2,a-2,-a+2,-a-2)$.

**Proof**

**of**

**Lemma**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

## 3. Unitary Similarity

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**5.**

**Theorem**

**5.**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**6.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | linear dichroism |

## References

- Hausdorff, F. Das Wertevorrat einer Bilinearform. Math. Z.
**1919**, 3, 314–316. [Google Scholar] [CrossRef] - Kippenhahn, R. Über den wertevorrat einer Matrix. Math. Nachr.
**1951**, 6, 193–228. [Google Scholar] [CrossRef] - Fiedler, M. Geometry of the numerical range of matrices. Linear Algebra Appl.
**1981**, 37, 81–96. [Google Scholar] [CrossRef] - Fiedler, M. Pencils of real symmetric matrices and real algebraic curves. Linear Algebra Appl.
**1990**, 141, 53–60. [Google Scholar] [CrossRef] - Lax, P.D. Differential equations, difference equations and matrix theory. Comm. Pure Appl. Math.
**1958**, 11, 175–194. [Google Scholar] [CrossRef] - Helton, J.W.; Vinnikov, V. Linear matrix inequality representations of sets. Commun. Pure Appl. Math.
**2007**, 60, 654–674. [Google Scholar] [CrossRef] - Helton, J.W.; Spitkovsky, I.M. The possible shapes of numerical ranges. Oper. Matrices
**2012**, 6, 607–611. [Google Scholar] [CrossRef] - Plaumann, D.; Sturmfels, B.; Vinzant, C. Computing linear matrix representations of Helton-Vinnikov curves. In Mathematical Methods in Systems, Optimization and Controls; Birkhauser: Basel, Switzerland, 2012; Volume 222, pp. 259–277. [Google Scholar]
- Chien, M.T.; Nakazato, H. Unitary similarity of the determinantal representation of unitary bordering matrices. Linear Algebra Appl.
**2018**, 541, 13–35. [Google Scholar] [CrossRef] - Chien, M.T.; Nakazato, H. Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials. Ann. Funct. Anal.
**2017**, 8, 152–167. [Google Scholar] [CrossRef] - Chien, M.T.; Nakazato, H. Determinantal representations of hyperbolic forms via weighted shift matrices. Appl. Math. Comput.
**2015**, 258, 172–181. [Google Scholar] [CrossRef] - Lentzos, K.; Pasley, L. Determinantal representations of invariant hyperbolic plane curves. arXiv, 2017; arXiv:1707.07724v1. [Google Scholar]
- Chien, M.T.; Liu, J.Z.; Nakazato, H.; Tam, T.Y. Toeplitz matrices are unitarily similar to symmetric matrices. Linear Multilinear Algebra
**2017**, 65, 2131–2144. [Google Scholar] [CrossRef] - Chien, M.T.; Nakazato, H. Boundary generating curves of c-numerical range. Linear Algebra Appl.
**1999**, 294, 67–84. [Google Scholar] [CrossRef] - Fröeberg, R. An Introduction to Gröbner Bases; John Wiley & Sons: New York, NY, USA, 1997. [Google Scholar]

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## Share and Cite

**MDPI and ACS Style**

Chien, M.-T.; Nakazato, H.
Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices ^{†}. *Symmetry* **2018**, *10*, 55.
https://doi.org/10.3390/sym10030055

**AMA Style**

Chien M-T, Nakazato H.
Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices ^{†}. *Symmetry*. 2018; 10(3):55.
https://doi.org/10.3390/sym10030055

**Chicago/Turabian Style**

Chien, Mao-Ting, and Hiroshi Nakazato.
2018. "Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices ^{†}" *Symmetry* 10, no. 3: 55.
https://doi.org/10.3390/sym10030055