Section 22.8

Complex Eigenvalues

When eigenvalues have imaginary parts, the solutions rotate.

Introduction

If , we get solutions with .

This combines growth/decay () with rotation ().

If eigenvector corresponds to :

The general solution is .

controls the "tightness" and direction of the spiral (in/out). controls the speed of rotation.

Solve .

1. Eigenvalues:

. ().

2. Eigenvector for +i:

3. Solution:

System with .

1. Classification:

Real part complex () Decay.

Imaginary part () Oscillation.

This is a Stable Spiral.

This matches Example 1.

Physics: No damping means energy is conserved.

Phase Portrait: Circles around origin (position vs velocity).