9.47(a), §2 Solution

Eigenvalues:

There are two roots: and

The eigenvector corresponding to satisfies

Solving using Gaussian elimination:

Equation (1) gives . In order to get a vector, instead of a set of possible vectors, one component must be arbitrarily chosen. Remember: undetermined constants in eigenvectors are not allowed. To get simple numbers, take v_1y = -2, then v_1x = 3:

Check:

Note: the null space of the matrix above is

so would also have been an acceptable eigenvector, just messier.

The eigenvector corresponding to satisfies

Solving using Gaussian elimination:

Choosing v_2y = 1, then v_2x = -2: