152 CHAPTER 6. SPECTRAL THEORY

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 thedirection  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 studiedin the form

x′′ = Ax (6.8)

where A is a real symmetric n × n matrix which has nonpositive eigenvalues. This isanalogous to the case of the scalar equation for undamped oscillation, x′′ + ω2x = 0. Themain difference is that here the scalar ω2 is replaced with the matrix −A. Consider theproblem of finding solutions to 6.8. You look for a solution which is in the form

x (t) = veλt (6.9)

and substitute this into 6.8. Thus

x′′ = vλ2eλt = eλtAv

and soλ2v = Av.

Therefore, λ2 needs to be an eigenvalue of A and v needs to be an eigenvector. Since Ahas nonpositive eigenvalues, λ2 = −a2 and so λ = ±ia where −a2 is an eigenvalue of A.Corresponding to this you obtain solutions of the form

x (t) = v cos (at) ,v sin (at) .

Note these solutions oscillate because of the cos (at) and sin (at) in the solutions. Here isan example.

Example 6.2.2 Find oscillatory solutions to the system of differential equations, x′′ = Axwhere

A =

 − 53 − 1

3 − 13

− 13 − 13

656

− 13

56 − 13

6

 .

The eigenvalues are −1,−2, and −3.

According to the above, you can find solutions by looking for the eigenvectors. Considerthe eigenvectors for −3. The augmented matrix for finding the eigenvectors is − 4

313

13 0

13 − 5

6 − 56 0

13 − 5

6 − 56 0

