6.2 Some Applications Of Eigenvalues And Eigenvectors
Recall that n×n matrices can be considered as linear transformations. If F is a 3 × 3 real matrix
having positive determinant, it can be shown that F = RU where R is a rotation matrix and U is
a symmetric real matrix having positive eigenvalues. An application of this wonderful
result, known to mathematicians as the right polar decomposition, is to continuum
mechanics where a chunk of material is identified with a set of points in three dimensional
space.
The linear transformation, F in this context is called the deformation gradient and it describes
the local deformation of the material. Thus it is possible to consider this deformation in terms of
two processes, one which distorts the material and the other which just rotates it. It is the matrix
U which is responsible for stretching and compressing. This is why in continuum mechanics, the
stress is often taken to depend on U which is known in this context as the right Cauchy Green
strain tensor. This process of writing a matrix as a product of two such matrices, one of which
preserves distance and the other which distorts is also important in applications to
geometric measure theory an interesting field of study in mathematics and to the study of
quadratic forms which occur in many applications such as statistics. Here I am emphasizing
the application to mechanics in which the eigenvectors of U determine the principle
directions, those directions in which the material is stretched or compressed to the maximum
extent.
Example 6.2.1Find the principle directions determined by the matrix
It is nice to be given the eigenvalues. The largest eigenvalue is 3 which means that
in the direction determined by the eigenvector associated with 3 the stretch is three
times as large. The smallest eigenvalue is 1∕2 and so in the direction determined by
the eigenvector for 1∕2 the material is compressed, becoming locally half as long. It
remains to find these directions. First consider the eigenvector for 3. It is necessary to
solve
and so the principle direction for the eigenvalue 3 in which the material is stretched to the
maximum extent is
( )
3
|( 1 |) .
1
A direction vector in this direction is
( √ -- )
3∕ 11
|( 1∕√11- |) .
1∕√11-
You should show that the direction in which the material is compressed the most is in the
direction
( )
| 0√- |
( − 1∕√ 2 )
1∕ 2
Note this is meaningful information which you would have a hard time finding without the
theory of eigenvectors and eigenvalues.
Another application is to the problem of finding solutions to systems of differential equations.
It turns out that vibrating systems involving masses and springs can be studied in the
form
′′
x = Ax (6.8)
(6.8)
where A is a real symmetric n×n matrix which has nonpositive eigenvalues. This is analogous to
the case of the scalar equation for undamped oscillation, x^{′′} + ω^{2}x = 0. The main difference is that
here the scalar ω^{2} is replaced with the matrix −A. Consider the problem of finding solutions to
6.8. You look for a solution which is in the form
Therefore, λ^{2} needs to be an eigenvalue of A and v needs to be an eigenvector. Since A has
nonpositive eigenvalues, λ^{2} = −a^{2} and so λ = ±ia where −a^{2} is an eigenvalue of A. Corresponding
to this you obtain solutions of the form
x (t) = v cos(at),vsin(at).
Note these solutions oscillate because of the cos
(at)
and sin
(at)
in the solutions. Here is an
example.
Example 6.2.2Find oscillatory solutions to the system of differential equations, x^{′′} = Axwhere
According to the above, you can find solutions by looking for the eigenvectors. Consider the
eigenvectors for −3. The augmented matrix for finding the eigenvectors is
are both solutions to the system of differential equations. You can find other oscillatory solutions
in the same way by considering the other eigenvalues. You might try checking these answers to
verify they work.
This is just a special case of a procedure used in differential equations to obtain closed form
solutions to systems of differential equations using linear algebra. The overall philosophy is to take
one of the easiest problems in analysis and change it into the eigenvalue problem which is the most
difficult problem in algebra. However, when it works, it gives precise solutions in terms of known
functions.