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(
T
xx
−
t
)(
T
yy
−
t
)
−
T
xy
T
yx
=
0
,
which gives the characteristic equation for the principal values
t
:
t
2
−
(
T
xx
+
T
yy
)
t
+
(
T
xx
T
yy
−
T
xy
T
yx
)
=
0
and the equation for the principal directions:
t
−
T
xx
T
xy
t
−
T
xx
+
T
yx
V
y
V
x
=
T
yx
tan
=
=
−
T
yy
=
+
T
xy
−
T
yy
,
(2
.
4)
t
t
where
is an angle between the vector
V
and the
x
-axis.
It follows from (2.3) and (2.4) that any symmetric real-valued tensor [
T
] has two
real principal values
(tr [
T
])
2
4
(
T
xx
+
T
yy
)
2
tr [
T
]
2
T
xx
+
T
yy
t
1
=
+
−
det [
T
]
=
+
−
(
T
xx
T
yy
−
T
xy
T
yx
)
,
2
4
(tr [
T
])
2
4
(
T
xx
+
T
yy
)
2
4
tr [
T
]
2
T
xx
+
T
yy
2
t
2
=
−
−
det [
T
]
=
−
−
(
T
xx
T
yy
−
T
xy
T
yx
)
(2
.
5)
and two orthogonal principal directions:
arctan
t
1
−
T
xx
+
T
yx
1
=
t
1
+
T
xy
−
T
yy
,
(2
.
6)
arctan
t
2
−
T
xx
+
T
yx
2
=
1
+
2
=
t
2
+
T
xy
−
T
yy
,
where tr [
T
]
T
xy
T
yx
.
From the solution of the eigenstate problem, we derive three independent
parameters,
t
1
,
=
T
xx
+
T
yy
and det [
T
]
=
T
xx
T
yy
−
t
2
and
1
, which fill all three degrees of freedom possessed by the
matrix [
T
].
Rotating the symmetric tensor [
T
] through the angle
1
, we reduce it to its prin-
cipal directions and get a diagonal tensor
t
1
0
[
T
]
=
.
(2
.
7)
0
t
2
Thus,
we
convey
all
the
information
inherent
in
the
four
components
T
xx
,
T
xy
,
T
yx
,
T
yy
of the tensor [
T
] to its principal values,
t
1
and
t
2
and principal
directions,
2.
On some alteration, this classical approach can be readily applied to the two-
dimensional impedance tensor. Really, with (1.62), that is, with
ske
1
,
2
=
1
+
/
w
=
0 and
S
