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and
V
H
,
real part of matrix is,
0.44717D+00 -0.19052D+00 -0.28493D+00
0.43326D+00 -0.84196D-01
-0.63165D+00
0.14267D+00 -0.31439D-02
0.30757D+00 -0.13213D+00
0.30071D+00 -0.24572D+00 -0.82148D-01
0.33106D+00 -0.29110D+00
-0.45811D+00 -0.22489D+00
0.85482D-01
0.52251D+00
0.12652D+00
0.31745D+00
0.46049D+00
0.59629D+00
0.44210D+00
0.31403D+00
imaginary part of matrix is,
0.00000D+00
0.40463D+00 -0.34475D+00 -0.11092D+00
0.43917D+00
0.00000D+00 -0.28401D+00
0.25425D-02 -0.80108D-01
0.61767D+00
0.00000D+00 -0.37157D+00
0.64460D+00 -0.29806D+00 -0.75420D-01
0.00000D+00
0.47903D+00
0.11350D+00 -0.83846D-01 -0.44029D+00
0.00000D+00 -0.91814D-01
0.43863D-01
0.15821D+00
0.46451D-01,
while the recovered matrix
C
=
U
·
W
·
V
H
is
real part of matrix is,
0.28488D+03 -0.12135D+03 -0.18145D+03 0.27585D+03 -0.53595D+02
-0.12135D+03 0.28488D+03 -0.12135D+03 -0.18145D+03 0.27585D+03
-0.18145D+03 -0.12135D+03 0.28488D+03 -0.12135D+03 -0.18145D+03
0.27585D+03 -0.18145D+03 -0.12135D+03
0.28488D+03 -0.12135D+03
-0.53595D+02
0.27585D+03 -0.18145D+03 -0.12135D+03
0.28488D+03
imaginary part of matrix is,
0.60215D-31 0.25772D+03 -0.21954D+03 -0.70624D+02 0.27956D+03
-0.25772D+03 0.15695D-13 0.25772D+03 -0.21954D+03 -0.70624D+02
0.21954D+03 -0.25772D+03 -0.69882D-13
0.25772D+03 -0.21954D+03
0.70624D+02
0.21954D+03 -0.25772D+03 -0.37131D-14
0.25772D+03
-0.27956D+03
0.70624D+02
0.21954D+03 -0.25772D+03
0.55629D-14.
Apart from rounding error in the diagonal elements of the imaginary part, the
reconstructed matrix is identical to the example matrix.
In the SVD factorisation (2.199) of the complex matrix
C
, the singular values are
required to be in descending order down the diagonal of the matrix
W
. To ensure
that this is the case, we sort the singular values and their associated eigenvectors.
If
u
1
,
u
2
,...,
u
M
are the column vectors of the matrix
U
and if
v
1
,
v
2
,...,
v
M
are
the column vectors of
V
, the product
U
V
H
may be expanded as
·
W
·
⎝
⎠
⎝
⎠
u
11
u
21
.
.
.
u
M
1
u
12
u
22
.
.
.
u
M
2
v
∗
11
v
∗
21
... v
∗
M
1
v
∗
12
v
∗
22
... v
∗
M
2
s
1
·
⊗
+
s
2
·
⊗
⎝
⎠
u
1
M
u
2
M
.
.
.
u
MM
v
∗
1
M
v
∗
2
M
... v
∗
MM
.
+ ···+
s
M
·
⊗
(2.243)
If, in the sorting, the singular values
s
i
and
s
j
are exchanged, then the
i
th and
j
th
column vectors of
U
, and the
i
th and
j
th row vectors of
V
H
, need to be exchanged.
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