Journal article
Minimal Realizations and Determinantal Representations in the Indefinite Setting
Integral equations and operator theory, v 94(2)
2022
Featured in Collection : UN Sustainable Development Goals @ Drexel
Abstract
For a signature matrix
J
, we show that a rational matrix function
M
(
z
) that is strictly
J
-contractive on the unit circle
T
, has a strict
J
~
⊕
J
-contractive realization
A
B
C
D
for an appropriate signature matrix
J
~
; that is,
M
(
z
)
=
D
+
z
C
(
I
-
z
A
)
-
1
B
. As an application, we use this result to show that a two variable polynomial
p
(
z
1
,
z
2
)
of degree
(
n
1
,
n
2
)
,
n
2
=
1
, without roots on
{
(
0
,
0
)
}
∪
(
T
×
{
0
}
)
∪
T
2
allows a determinantal representation
1
p
(
z
1
,
z
2
)
=
p
(
0
,
0
)
det
(
I
n
1
+
1
-
K
Z
)
,
Z
=
z
1
I
n
1
⊕
z
2
I
n
2
,
where
K
is a strict
J
~
⊕
J
-contraction. This provides first evidence of a new conjecture that a two variable polynomial
p
(
z
1
,
z
2
)
of degree
(
n
1
,
n
2
)
has a determinantal representation (
1
) with
K
a strict
J
~
⊕
J
-contraction if and only if
p
(
z
1
,
z
2
)
has no roots in
{
(
0
,
0
)
}
∪
T
2
.
Metrics
Details
- Title
- Minimal Realizations and Determinantal Representations in the Indefinite Setting
- Creators
- Joshua D. Jackson - Drexel University College of Arts and SciencesHugo J. Woerdeman - Drexel University
- Publication Details
- Integral equations and operator theory, v 94(2)
- Publisher
- Springer International Publishing
- Grant note
- 2000037 / directorate for mathematical and physical sciences (http://dx.doi.org/10.13039/100000086) 355645 / simons foundation (http://dx.doi.org/10.13039/100000893)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000791788300001
- Scopus ID
- 2-s2.0-85129616135
- Other Identifier
- 991019168798804721
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