has
three distinct roots.
In
elasticity stress tensor, strain tensor, rate of deformation tensor are all real
and symmetric. Eigen values of any real symmetric tensor are all real. So, there
are at least three eigen vectors called Principal
directions, and corresponding
eigen values called Principal values. Also in this special case, the principal
directions are mutually perpendicular.
Case
(i)
If the characteristic equation
has
three distinct roots.
There will be three distinct eigen values and three eigen
vectors mutually perpendicular to each other.
Case
(ii) If
the characteristic equation has one distinct root
and
two repeated roots
There will be two distinct eigen values. There will be an
eigen vector corresponding to
. Any line vertical to
will also be an eigen vector. Or any vector lying in the
plane vertical to
will be
an eigen vector.
Case
(iii)
All the three roots are equal.
There is only one eigen value. Any vector in the domain
is an eigen vector.
(e.g.)
tensor
Corresponding
to the three mutually vertical directions of eigen vectors , the tensor
can be
transformed by coordinate transformation with
, as the unit base vectors
to
give a diagonal matrix with eigen values.
With
the convention
