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2
C
12
2
xy
x
2
tan2
ϕ
=
C
11
− C
22
=
− y
2
.
(2.11)
If
x
=
y,
then tan
ϕ
=
−
1
,
tan 2
ϕ →±∞,ϕ
=
∓
45
◦
.
End of Example.
2-2 Eigenspace Analysis, Euler-Lagrange Deformation Tensor
Left and right eigenspace analysis and synthesis of the Euler-Lagrange deformation tensor,
special case
{
M
r
,g
μν
}
=
{
R
2
,δ
μν
}
.
First, let us confront you with Lemma
2.2
, where we present detailed results of the left and right
eigenspace analysis and synthesis of the Euler-Lagrange deformation tensor for the special case
of a right Euclidean manifold. Second, we focus on an interpretation of the results.
Lemma 2.2 (Left and right eigenspace analysis and synthesis of the Euler-Lagrange deformation
tensor, special case
2
2
,δ
μν
}
{
M
r
,g
μν
}
=
{
R
).
(i) Synthesis.
For the pair of symmetric matrices
are posi-
tive definite, a simultaneous diagonalization is (the right Frobenius matrix F
r
is an orthonormal
matrix)
{
E
l
,
G
l
}
or
{
E
r
,
G
r
}
,
where the matrices
{
G
l
,
G
r
}
F
l
E
l
F
l
=diag[
K
1
,K
2
]
,
F
l
G
i
F
l
=I versus
F
r
E
r
F
r
=diag[
κ
1
,κ
2
]
,
F
r
F
1
=I
.
(2.12)
(ii) Analysis.
Left eigenvalues:
|
E
l
−
K
i
G
l
|
=0
, K
1
,
2
=
K
±
=
1
2
tr[E
l
G
−
l
]
±
−
4det [E
l
G
−
l
]
.
(tr[E
l
G
−
l
])
2
(2.13)
Left eigencolumns:
F
11
F
21
=
1
G
11
(
e
22
K
1
G
12
)
2
×
−
K
1
G
22
)
2
−
2
G
12
(
e
12
−
K
1
G
12
)(
e
22
−
K
1
G
22
)+
G
22
(
e
12
−
e
22
−
,
K
1
G
22
×
(2.14)
−
(
e
12
−
K
1
G
12
)
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