268 CHAPTER 12. SPECTRAL THEORY
positive. The symmetry of U allows the proof of a theorem which says that if λ M is thelargest eigenvalue, then in every other direction other than the one corresponding to theeigenvector for λ M the material is stretched less than λ M and if λ m is the smallest eigen-value, then in every other direction other than the one corresponding to an eigenvector ofλ m the material is stretched more than λ m. This process of writing a matrix as a prod-uct of two such matrices, one of which preserves distance and the other which distorts isalso important in applications to geometric measure theory an interesting field of study inmathematics and to the study of quadratic forms which occur in many applications such asstatistics. Here we are emphasizing the application to mechanics in which the eigenvectorsof the symmetric matrix U determine the principal directions, those directions in whichthe material is stretched the most or the least.
Example 12.2.1 Find the principal directions determined by the matrix
2911
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 in thedirection 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 for1/2 the material is stretched by a factor of 1/2, becoming locally half as long. It remainsto find these directions. First consider the eigenvector for 3. It is necessary to solve3
1 0 00 1 00 0 1
−
2911
611
611
611
4144
1944
611
1944
4144
x
yz
=
000
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 00 0 0 0