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we need
V
=
U
∗
. So how exactly is the Takagi factorization determined from the SVD?
Because the matrix
C
is symmetric, it has two SVDs:
C
=
UKV
H
=
V
∗
KU
T
=
C
T
.
(3.36)
Now, since
CC
H
UK
2
U
H
V
∗
K
2
V
T
K
2
U
H
V
∗
=
U
H
V
∗
K
2
=
=
⇔
⇔
KU
H
V
∗
=
U
H
V
∗
K
,
(3.37)
the unitary matrix
U
H
V
∗
commutes with
K
. This is possible only if the (
i
,
j
)th element
of
U
H
V
∗
is zero whenever
k
i
=
k
j
.Thus,every
C
can be expressed as
C
=
UKDU
T
(3.38)
n
with
D
=
V
H
U
∗
. Assume that among the circularity coefficients
{
k
i
}
i
=
1
there are
N
dis-
tinct coefficients, denoted by
σ
1
,...,σ
N
. Their respective multiplicities are
m
1
,...,
m
N
.
Then we may write
D
=
Diag
(
D
1
,
D
2
,...,
D
N
)
,
(3.39)
where
D
i
is an
m
i
×
m
i
unitary and symmetric matrix, and
=
σ
1
D
1
,σ
2
D
2
,...,σ
N
D
N
)
.
KD
Diag
(
(3.40)
In the special case in which all circularity coefficients are distinct, we have
σ
i
=
k
i
and
D
i
is a scalar with unit magnitude. Therefore,
Diag
(e
j
θ
1
e
j
θ
2
e
j
θ
n
)
D
=
,
,...,
(3.41)
Diag
(
k
1
e
j
θ
1
k
2
e
j
θ
2
k
n
e
j
θ
n
)
=
,
,...,
.
KD
(3.42)
Then we have
UD
1
/
2
KD
T
/
2
U
T
C
=
(3.43)
with
D
1
/
2
e
j
θ
1
/
2
e
j
θ
2
/
2
e
j
θ
n
/
2
)
=
Diag
(
±
,
±
,...,
±
(3.44)
FKF
T
.
This shows how to obtain the Takagi factorization from the SVD. We note that, if only
the circularity coefficients
k
i
need to be computed and the canonical coordinates
UD
1
/
2
with arbitrary signs, so that
F
=
achieves the Takagi factorization
C
=
are
not required, a regular SVD will suffice.
The SVD
C
UKV
H
of any matrix
C
with distinct singular values is unique up to
multiplication of
U
from the right by a unitary diagonal matrix. If
C
=
C
T
, the Takagi
factorization determines this unitary diagonal factor in such a way that the matrix of
left singular vectors is the conjugate of the matrix of right singular vectors. Hence, if all
circularity coefficients are distinct, the Takagi factorization is unique up to the sign of
the diagonal elements in
D
1
/
2
. The following result then follows.
=
Result 3.3.
For distinct circularity coefficients, the strong uncorrelating transform
A
=
F
H
R
−
1
/
2
xx
is unique up to the sign of its rows,
regardless
of the choice of the inverse
square root of
R
xx
.
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