Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †
Let A be an n × n complex matrix. Assume the determinantal curve V A...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2018-02-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | http://www.mdpi.com/2073-8994/10/3/55 |
| _version_ | 1811184731137507328 |
|---|---|
| author | Mao-Ting Chien Hiroshi Nakazato |
| author_facet | Mao-Ting Chien Hiroshi Nakazato |
| author_sort | Mao-Ting Chien |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T13:18:26Z |
| format | Article |
| id | doaj.art-bed42f50e4fb4e0a99aa1c79ed9ee8ce |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T13:18:26Z |
| publishDate | 2018-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-bed42f50e4fb4e0a99aa1c79ed9ee8ce2022-12-22T04:22:20ZengMDPI AGSymmetry2073-89942018-02-011035510.3390/sym10030055sym10030055Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices †Mao-Ting Chien0Hiroshi Nakazato1Department of Mathematics, Soochow University, Taipei 11102, TaiwanFaculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, JapanLet 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.http://www.mdpi.com/2073-8994/10/3/55determinantal representationhyperbolic ternary formsrational curvestoeplitz matricesnumerical range |
| spellingShingle | Mao-Ting Chien Hiroshi Nakazato Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices † Symmetry determinantal representation hyperbolic ternary forms rational curves toeplitz matrices numerical range |
| title | Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices † |
| title_full | Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices † |
| title_fullStr | Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices † |
| title_full_unstemmed | Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices † |
| title_short | Symmetric Representation of Ternary Forms Associated to Some Toeplitz Matrices † |
| title_sort | symmetric representation of ternary forms associated to some toeplitz matrices † |
| topic | determinantal representation hyperbolic ternary forms rational curves toeplitz matrices numerical range |
| url | http://www.mdpi.com/2073-8994/10/3/55 |
| work_keys_str_mv | AT maotingchien symmetricrepresentationofternaryformsassociatedtosometoeplitzmatrices AT hiroshinakazato symmetricrepresentationofternaryformsassociatedtosometoeplitzmatrices |