Real Eigenvalues

Main idea

With two different real eigenvalues, every solution is a mix of two straight-line motions, one along each eigenvector, growing or shrinking at its own rate. Over time the faster-growing one takes over, so trajectories end up running parallel to its eigenvector.

\vec x(t) = {\htmlData{sym=0}{c_1}}e^{\lambda_1t}{\htmlData{sym=1}{\vec v_1}} + c_2e^{\lambda_2t}\vec v_2

What the symbols mean

$c_1$
First constant
With c₂, chosen so that x(0) = c₁v₁ + c₂v₂ matches the initial condition.
$\vec v_1$
Eigenvector
The direction this term moves along, growing or shrinking at rate λ₁.

Explanation

Real eigenvalues are simply stretch factors that are ordinary real numbers (no $i$ involved), so nothing in the solution rotates. All we're really saying is that the motion then splits into two straight-line motions added together, which makes these systems the easiest to solve and picture. The formula $\vec x(t) = c_1e^{\lambda_1t}\vec v_1 + c_2e^{\lambda_2t}\vec v_2$ looks long, but each half is the same three pieces: $\vec v_1$ (a direction), $e^{\lambda_1t}$ (how it grows or shrinks over time), and $c_1$ (how much of it you start with). Add the two halves and you have every solution. It's just two easy pieces glued together.

Why this is every solution. From the eigenvalues lesson, each $e^{\lambda t}\vec v$ solves $\vec x' = A\vec x$ on its own. The system is linear, so any combination of solutions is a solution too, just like $c_1e^{r_1t} + c_2e^{r_2t}$ for a second-order equation. When $\lambda_1 \ne \lambda_2$ the two eigenvectors point in different directions, so $c_1\vec v_1 + c_2\vec v_2$ can hit any starting point $\vec x(0)$ you like. That's enough to match every initial condition, so it's the general solution.

Steps:

Find the eigenvalues and an eigenvector for each.
Write the general solution.
Solve $c_1\vec v_1 + c_2\vec v_2 = \vec x(0)$ for the constants.

Long-term behavior: the term with the larger eigenvalue dominates, so trajectories end up parallel to its eigenvector. Think about why: $e^{4t}$ outgrows $e^{-t}$ enormously as $t$ increases, so after a while the other term is too small to see.

A common mistake. It's tempting to think the term with the bigger constant $c$ is the one that wins in the long run, because it starts out larger. It isn't: the exponent decides. A tiny $c_1$ times $e^{4t}$ still swamps a big $c_2$ times $e^{-t}$ after a short while. The only way to avoid the dominant direction is to have its constant be exactly $0$ .

See it

The system x′ = [[1, 2], [3, 2]]x, with eigenvalues 4 and −1. Along v₂ = (1, −1) solutions shrink toward the origin, along v₁ = (2, 3) they race away, and everything else does a bit of both. The moving point, which passes through (0, −4) at t = 0, comes in close to v₂ and then gets flung out parallel to v₁. That's a saddle.

Worked examples

Example 1

Solve $\vec x' = \begin{pmatrix} 1 & 2\\ 3 & 2\end{pmatrix}\vec x$ , $\vec x(0) = \binom{0}{-4}$ .

1
The trace is $3$ and the determinant is $-4$ , so $\lambda^2 - 3\lambda - 4 = 0$ and $\lambda = 4, -1$ . Their eigenvectors are $\binom23$ and $\binom{1}{-1}$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the eigenvalues of $\begin{pmatrix} 1 & 2\\ 3 & 2\end{pmatrix}$ .

Problem 2+5 XP

Give the eigenvector for $\lambda = 4$ with first component 2.

Problem 3+5 XP

For $\vec x(0) = \binom{0}{-4}$ , find $c_1, c_2$ .

Problem 4+5 XP

Write $x_1(t)$ for that solution.

Problem 5+5 XP

What kind of equilibrium is the origin here?

Problem 6+5 XP

A drug moves from the gut into the blood: $x_1' = -2x_1$ (gut) and $x_2' = 2x_1 - x_2$ (blood), in mg. With $x_1(0) = 10$ and $x_2(0) = 0$ , find $x_2(t)$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Solve $\vec x\,' = \begin{pmatrix} 3 & -2\\ 2 & -2\end{pmatrix}\vec x$ with $\vec x(0) = \binom13$ , and give $x_2(t)$ .

Where it's used

Two tanks exchanging brine, two connected compartments in a drug model, or two competing species near coexistence: real-eigenvalue systems.

A Pill's Path from Gut to BloodDiff Eq · Medicine

The Phase Plane Complex Eigenvalues