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In-Depth Information
This decomposition of the impedance matrix differs from the standard singular
value decomposition, SVD, in that the diagonal elements
] are com-
plex. Let us keep the standard SVD terminology and call these values the complex
singular values. In the La Torraca-Madden-Korringa method the complex singular
values,
1
and
2
of [
2
, are considered as principal values of the impedance tensor.
Matrix equation (2.76) has eight unknowns quantities (polarization parameters,
E
1
,
E
1
,
1
and
H
1
,
H
1
, and complex singular values,
1
and
2
) against eight known quan-
tities (complex components,
Z
xx
,
Z
yy
).
Moduli of complex singular values and polarization parameters are readily
derived by standard SVD procedure. Introduce matrices
Z
xy
,
Z
yx
,
+
Z
xy
Z
xx
Z
xy
Z
yx
Z
yy
Z
xx
2
Z
yx
Z
xx
Z
yx
+
Z
xy
Z
yy
2
|
Z
xx
|
[
C
E
]
=
[
Z
] [
Z
]
=
=
Z
xy
Z
yy
Z
yx
+
Z
yy
Z
xy
Z
yy
Z
xx
Z
yx
+
2
2
Z
xx
Z
xx
Z
xy
Z
yx
Z
yy
+
Z
yx
2
Z
yx
Z
xx
Z
xy
+
Z
yx
Z
yy
|
Z
xx
|
2
[
C
H
]
=
[
Z
][
Z
]
=
=
Z
xx
Z
xy
+
Z
yx
Z
yy
Z
xy
+
Z
yy
,
Z
xy
Z
yy
2
2
(2
.
77)
where [Z]
=
[
U
e
][
] [
U
h
] and [Z]
=
[
U
h
][
] [
U
e
]. Here
+
Z
xy
+
Z
yx
+
Z
yy
2
2
2
2
2
tr [
C
E
]
=
tr [
C
H
]
= |
Z
xx
|
=
Z
,
(2
.
78)
=
Z
xx
Z
yy
−
Z
xy
Z
yx
2
2
det [
C
E
]
=
det [
C
H
]
= |
det [
Z
]
|
,
where
.
It is seen that [
C
E
] and [
C
H
] are Hermitian matrices: their principal diagonal
elements are real, while the secondary diagonal elements are complex conjugate.
The remarkable feature of Hermitian matrices is that their eigenvalues are real. In
fact,
Z
and det [
Z
] are the Euclidean norm and determinant of the matrix [
Z
]
[
C
E
]
=
[
U
e
][
] [
U
h
][
U
h
][
] [
U
e
]
=
[
U
e
][
] [
U
e
]
,
[
C
H
]
=
[
U
h
][
] [
U
e
][
U
e
][
] [
U
h
]
=
[
U
h
][
] [
U
h
]
,
whence
[
C
E
][
U
e
]
=
[
U
e
]
[
]
,
(2
.
79)
[
C
H
][
U
h
]
=
[
U
h
][
S
]
,
where
|
1
|
=
2
0
.
2
0
|
2
|