Section 22.7

Real Distinct Eigenvalues

When the matrix stretches space along two independent lines. The simplest and most common case.

Introduction

If are both real, we have two independent eigenvectors .

.
All solutions fly away to infinity.
They leave tangent to the "slow" eigenvector () and parallel to the "fast" one ().

.
All solutions die out to zero.
They approach tangent to the "slow" eigenvector (closest to 0).

If .

One direction pulls in (), one pushes out ().
Most trajectories curve: they come in along , turn, and go out along .

Solve .

1. Eigenvalues:

. Saddle.

2. Eigenvectors:

For : .

3. Solution:

Solve .

1. Check Eigenvalues:

2. Eigenvectors:

Standard process yields .

Both negative eigenvalues Sink.

Sketch the phase portrait for Example 1.

Draw line (from ). Arrows point OUT (exp growth).

Draw line (from ). Arrows point IN (exp decay).

Other curves follow the IN line initially and bend towards the OUT line.