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 |
| Summary: | 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. |
|---|---|
| ISSN: | 2073-8994 |