Section 22.9

Repeated Eigenvalues

What happens when we run out of eigenvectors? We need a "backup plan": Generalized Eigenvectors.

Introduction

Sometimes gives only one eigenvector.

We know one solution .
But we need a second one! Remember reduction of order? It involved multiplying by . Same idea here.

The second solution is:

Where satisfies:
.

Also known as a "Defective Node". All trajectories eventually become parallel to the single eigenvector.

Solve .

1. Eigenvalues:

2. Eigenvector v:

3. Generalized vector w:

Let , then . .

Construct the general solution for Example 1.

Is the origin stable or unstable?

The solution grows exponentially.

Unstable Improper Node.