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This matrix equation breaks down into four equations, which conform to classical
eigenstate formulation (2.2):
2
e
1
2
e
2
[
C
E
]
e
1
= |
1
|
[
C
E
]
e
2
= |
2
|
(2
.
80)
2
h
1
2
h
2
.
[
C
H
]
h
1
= |
1
|
[
C
H
]
h
2
= |
2
|
2
2
|
1
|
,
|
2
|
It is apparent that
are eigenvalues of matrices [
C
E
]
,
[
C
H
], while
e
1
,
h
2
are their eigenvectors.
In line with (2.5) and (2.78), we get
e
2
and
h
1
,
J
11
−
2
J
11
+
|
I
2
|
4
2
4
2
=
Z
+
Z
−
4
|
det [
Z
]
|
2
|
1
|
=
2
2
(2
.
81)
J
11
−
J
11
−
2
4
|
I
2
|
2
4
2
=
|
|
Z
−
Z
−
4
det [
Z
]
2
|
2
|
=
,
2
2
where
1
and
2
are the complex principal values of the impedance tensor [
Z
]
.
Mo-
duli of
1
and
2
are expressed in ter
ms of rotational invariants I
2
=
det [Z]
=
+
Z
xy
+
Z
yx
+
Z
yy
2
2
2
.Theyare
2
Z
xx
Z
yy
−
Z
xy
Z
yx
and
J
11
=
Z
=
|
Z
xx
|
independent of the orientation of measurement axes. Note that
|
1
| ≥ |
2
|
.
Similarly, in the line with (2.4) (2.69) and (2.77), we get
−
Z
xy
2
Z
xx
Z
yx
+
Z
xy
Z
yy
2
2
P
E
1
=
|
1
|
− |
Z
xx
|
=
−
Z
yx
−
Z
yy
Z
xx
Z
yx
+
Z
xy
Z
yy
2
2
2
|
1
|
−
Z
xy
2
Z
xx
Z
yx
+
Z
xy
Z
yy
2
2
|
1
|
− |
Z
xx
|
+
=
Z
xy
Z
yy
,
Z
yx
Z
yy
2
2
Z
xx
Z
yx
+
2
|
1
|
−
−
+
(2
.
82)
Z
yx
2
2
2
Z
xx
Z
xy
+
Z
yx
Z
yy
P
H
1
=
|
1
|
− |
Z
xx
|
−
=
Z
xy
Z
yy
Z
xx
Z
xy
+
Z
yx
Z
yy
2
2
2
|
1
|
−
−
Z
yx
2
Z
xx
Z
xy
+
Z
yx
Z
yy
2
2
|
1
|
−
|
Z
xx
|
−
+
=
Z
yx
Z
yy
.
Z
xy
Z
yy
2
2
Z
xx
Z
xy
+
2
|
1
|
−
−
+
Knowing
P
E
1
and
P
H
1
, we use (2.62), (2.63) and taking into account (2.68) derive
principal directions,
E
1
and
H
1
, as well as ellipticities,
E
1
and
H
1
.
