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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 |
 
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