Page 53 - Introduction to Continuum Mechanics

38 Tensors

T
r
r
Thus, (R R)m = R m. Since R R = I, we have

Thus, (i) and (ii) gives

4
But (R-R )m = 2qXm, where q is the dual vector of R . Thus,

i.e., q is parallel to m. We note that it can be shown (see Prob. 2B29 or Prob. 2B36) that if &
denotes the right-hand rotation angle, then

2B17 Eigenvalues and Eigenvectors of a Tensor

Consider a tensor T. If a is a vector which transforms under T into a vector parallel to itself,
i.e.,

then a is an eigenvector and A is the corresponding eigenvalue.
If a is an eigenvector with corresponding eigenvalue A of the linear transformation T, then
any vector parallel to a is also an eigenvector with the same eigenvalue A. In fact, for any scalar
a,

Thus, an eigenvector, as defined by Eq. (2B17.1), has an arbitrary length. For definiteness, we
shall agree that all eigenvectors sought will be of unit length.
A tensor may have infinitely many eigenvectors. In fact, since la = a, any vector is an
eigenvector for the identity tensor I, with eigenvalues all equal to unity. For the tensor /?!, the
same is true, except that the eigenvalues are all equal toft.
Some tensors have eigenvectors in only one direction. For example, for any rotation tensor,
which effects a rigid body rotation about an axis through an angle not equal to integral multiples
of jc, only those vectors which are parallel to the axis of rotation will remain parallel to
themselves.
Let n be a unit eigenvector, then

Thus,