Section 1.11
Solid Mechanics Part III
Kelly
96
1.11 The Eigenvalue Problem and Polar Decomposition
1.11.1
Eigenvalues, Eigenvectors and Invariants of a Tensor
Consider a second-order tensor
A
.
Suppose that one can find a scalar
λ
and a (non-zero)
normalised, i.e. unit, vector
n
ˆ
such that
n
n
A
ˆ
ˆ
λ
=
(1.11.1)
In other words,
A
transforms the vector
n
ˆ
into a vector parallel to itself, Fig. 1.11.1.
If
this transformation is possible, the scalars are called the
eigenvalues
(or
principal
values
) of the tensor, and the vectors are called the
eigenvectors
(or
principal directions
or
principal axes
) of the tensor.
It will be seen that there are
three
vectors
n
ˆ
(to each of
which corresponds some scalar
λ
) for which the above holds.
Figure 1.11.1: the action of a tensor A on a unit vector
Equation 1.11.1 can be solved for the eigenvalues and eigenvectors by rewriting it as
(
)
0
ˆ
=
−
n
I
A
λ
(1.11.2)
or, in terms of a Cartesian coordinate system,
(
)
(
)
(
)
0
ˆ
ˆ
0
ˆ
ˆ
0
ˆ
ˆ
=
−
→
=
−
→
=
⊗
−
⊗
i
i
j
ij
r
r
i
j
ij
r
r
q
p
pq
k
k
j
i
ij
n
n
A
n
n
A
n
n
A
e
e
e
e
e
e
e
e
e
λ
λ
λδ
In full,
[
]
[
]
[
]
0
ˆ
)
(
ˆ
ˆ
0
ˆ
ˆ
)
(
ˆ
0
ˆ
ˆ
ˆ
)
(
3
3
33
2
32
1
31
2
3
23
2
22
1
21
1
3
13
2
12
1
11
=
−
+
+
=
+
−
+
=
+
+
−
e
e
e
n
A
n
A
n
A
n
A
n
A
n
A
n
A
n
A
n
A
λ
λ
λ
(1.11.3)
Dividing out the base vectors, this is a set of three homogeneous equations in three
unknowns (if one treats
λ
as known).
From basic linear algebra, this system has a
solution (apart from
0
ˆ
=
i
n
) if and only if the determinant of the coefficient matrix is
zero, i.e. if
A
n
ˆ
n
ˆ
λ

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Section 1.11
Solid Mechanics Part III
Kelly
97
0
det
)
det(
33
32
31
23
22
21
13
12
11
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
=
−
λ
λ
λ
λ
A
A
A
A
A
A
A
A
A
I
A
(1.11.4)
Evaluating the determinant, one has the following cubic
characteristic equation
of
A
,
0
III
II
I
2
3
=
−
+
−
A
A
A
λ
λ
λ
Tensor Characteristic Equation
(1.11.5)
where
(
)
[
]
A
A
A
A
A
A
A
det
III
)
tr(
)
(tr
II
tr
I
3
2
1
2
2
2
1
2
1
=
=
−
=
−
=
=
=
k
j
i
ijk
ij
ji
jj
ii
ii
A
A
A
A
A
A
A
A
ε
(1.11.6)
It can be seen that there are three roots
3
2
1