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of two positive-definite symmetric matrices
{
C
l
,
G
l
}
or
{
C
r
,
G
r
}
, respectively. Compare with
Lemma
1.5
.
Lemma 1.5 (Left and right general eigenvalue problem of the Cauchy-Green deformation tensor).
For the pair of positive-definite symmetric matrices
{
C
l
,
G
l
}
or
{
C
r
,
G
r
}
, respectively, a simulta-
neous diagonalization defined by
left diagonalization:
right diagonalization:
F
l
C
l
F
l
=diag(
Λ
1
,Λ
2
):=D
l
,
F
r
C
r
F
r
=diag(
λ
1
,λ
2
):=D
r
,
(1.56)
versus
F
l
G
l
F
l
=I
2
F
r
G
I
F
r
=I
2
is readily obtained from the following general eigenvalue-eigenvector problem of type
left eigen-
values
and
left principal stretches
:
C
l
F
l
−
G
l
F
l
D
l
=0
⇐⇒
Λ
i
G
l
)
f
li
=0
⇐⇒
(C
l
−
(1.57)
Λ
2
G
l
|
|
C
l
−
=0
,
tr[C
l
G
−
l
]
4det[C
l
G
−
l
]
,
(tr[C
l
G
−
l
])
2
=
1
2
Λ
1
,
2
=
Λ
2
±
−
±
subject to F
l
G
l
F
l
=I
2
,
and
C
r
F
r
−
G
r
F
r
D
r
=0
⇐⇒
λ
i
G
r
)
f
r
i
=0
⇐⇒
(C
r
−
(1.58)
λ
2
G
r
|
C
r
−
|
=0
,
tr[C
r
G
−
r
]
4det[C
r
G
−
r
]
,
(tr[C
r
G
−
r
])
2
=
1
2
λ
1
,
2
=
λ
2
±
−
±
subject to F
r
G
r
F
r
=I
2
,
and
Λ
1
,
2
=1
/λ
1
,
2
⇐⇒
1
/Λ
1
,
2
=
λ
1
,
2
.
(1.59)
End of Lemma.
In order to visualize the eigenspace of the left and right Cauchy-Green deformation tensors
C
l
and C
r
relative to the left and right metric tensors G
i
and G
r
, we are forced to compute in
addition the
eigenvectors
, in particular, the
eigencolumns
(also called
eigendirections
) of the pairs
{
C
l
,
G
l
}
and
{
C
r
,
G
r
}
, respectively. Compare with Lemma
1.6
.
Lemma 1.6 (Left and right general eigenvectors, left and right principal stretch directions).
For the pair of positive-definite symmetric matrices
, an explicit form of the
left eigencolumns (also called
left principal stretch directions
) and of the right eigencolumns (also
called
right principal stretch directions
)is
{
C
l
,
G
l
}
and
{
C
r
,
G
r
}
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