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E
xm
=
m
H
ym
,
E
ym
=−
m
H
xm
,
m
=
1
,
2
,
(2
.
41)
where
ς
m
is the complex principal value (eigenvalue, principal impedance) of the
impedance tensor [
Z
]
.
Thus,
=
m
01
−
H
τ
m
,
E
τ
m
=
[
Z
]
H
τ
m
m
=
1
,
2
.
(2
.
42)
10
Rotating the magnetic eigenfield
H
τ
m
and the impedance [
Z
] through
±
/
2, we
obtain the Adam formulation (1.16) of the eigenstate problem
[
Z
]
H
τ
m
=
m
H
τ
m
,
m
=
1
,
2
,
(2
.
43)
which coincides with the standard formulation (2.2). Here, with account for (1.1 6),
H
y
−
H
τ
=
[
R
(
/
2)]
H
τ
=
H
x
Z
xy
−
Z
xx
[
Z
]
=
[
Z
][
R
(
−
/
2)]
=
.
Z
yy
−
Z
yx
Writing (2.43) in full, we get
Z
xx
H
xm
+
(
Z
xy
−
m
)
H
ym
=
0
,
(2
.
44)
(
Z
yx
+
m
)
H
xm
+
Z
yy
H
ym
=
0
,
m
=
1
,
2
.
Let the determinant of this uniform system be zero. Then
2
m
−
(
Z
xy
−
Z
yx
)
m
+
(
Z
xx
Z
yy
−
Z
xy
Z
yx
)
=
0
,
m
=
1
,
2
.
(2
.
45)
Solving this characteristic equation, we find the complex principal values (eigen-
values) of the tensor [
Z
]:
(
Z
xy
−
Z
yx
)
2
I
3
−
I
3
+
4I
2
Z
xy
−
Z
yx
+
−
4(
Z
xx
Z
yy
−
Z
xy
Z
yx
)
1
2
1
=
=
2
(
Z
xy
−
Z
yx
)
2
I
3
−
I
3
−
4I
2
Z
xy
−
Z
yx
−
−
4(
Z
xx
Z
yy
−
Z
xy
Z
yx
)
1
2
2
=
=
2
(2
.
46)
tr
Z
expressed in terms of rotational invariants I
3
=
=
Z
xy
−
Z
yx
and I
2
=
=
Z
xx
Z
yy
−
Z
xy
Z
yx
.
det [
Z
]
