6.2. SOME APPLICATIONS OF EIGENVALUES AND EIGENVECTORS 151
matrix and U is a symmetric real matrix having positive eigenvalues. An application ofthis wonderful result, known to mathematicians as the right polar decomposition, is tocontinuum mechanics where a chunk of material is identified with a set of points in threedimensional space.
The linear transformation, F in this context is called the deformation gradient andit describes the local deformation of the material. Thus it is possible to consider thisdeformation in terms of two processes, one which distorts the material and the other whichjust rotates it. It is the matrix U which is responsible for stretching and compressing. Thisis why in continuum mechanics, the stress is often taken to depend on U which is known inthis context as the right Cauchy Green strain tensor. This process of writing a matrix as aproduct of two such matrices, one of which preserves distance and the other which distortsis also important in applications to geometric measure theory an interesting field of studyin mathematics and to the study of quadratic forms which occur in many applications suchas statistics. Here I am emphasizing the application to mechanics in which the eigenvectorsof U determine the principle directions, those directions in which the material is stretchedor compressed to the maximum extent.
Example 6.2.1 Find the principle directions determined by the matrix2911
611
611
611
4144
1944
611
1944
4144
The eigenvalues are 3, 1, and 1
2 .
It is nice to be given the eigenvalues. The largest eigenvalue is 3 which means that inthe direction determined by the eigenvector associated with 3 the stretch is three times aslarge. The smallest eigenvalue is 1/2 and so in the direction determined by the eigenvectorfor 1/2 the material is compressed, becoming locally half as long. It remains to find thesedirections. First consider the eigenvector for 3. It is necessary to solve3
1 0 0
0 1 0
0 0 1
−
2911
611
611
611
4144
1944
611
1944
4144
x
y
z
=
0
0
0
Thus the augmented matrix for this system of equations is
411 − 6
11 − 611 0
− 611
9144 − 19
44 0
− 611 − 19
449144 0
The row reduced echelon form is 1 0 −3 0
0 1 −1 0
0 0 0 0
and so the principle direction for the eigenvalue 3 in which the material is stretched to themaximum extent is 3
1
1
.