14Tensors of rank 2
IA Vector Calculus
14.3 Diagonalization of a symmetric second rank tensor
Recall that using matrix notation,
T = (T
ij
), T
0
= (T
0
ij
), R = (R
ij
),
and the tensor transformation rule T
0
ij
= R
ip
R
jq
T
pq
becomes
T
0
= RT R
T
= RT R
−1
.
If
T
is symmetric, it can be diagonalized by such an orthogonal transformation.
This means that there exists a basis of orthonormal eigenvectors
e
1
, e
2
, e
3
for
T
with real eigenvalues
λ
1
, λ
2
, λ
3
respectively. The directions defined by
e
1
, e
2
, e
3
are the principal axes for
T
, and the tensor is diagonal in Cartesian coordinates
along these axes.
This applies to any symmetric rank-2 tensor. For the special case of the
inertia tensor, the eigenvalues are called the principal moments of inertia.
As exemplified in the previous example, we can often guess the correct
principal axes for
I
ij
based on the symmetries of the body. With the axes we
chose, I
ij
was found to be diagonal by direct calculation.