Eigenvalues and Eigenvectors

Main idea

An eigenvector of $A$ is a direction that $A$ doesn't turn, only stretches, and the eigenvalue $\lambda$ is the stretch factor. You find the eigenvalues as the roots of $\det(A - \lambda I) = 0$ , and they matter because along those directions a system just grows or shrinks like $e^{\lambda t}$ .

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What the symbols mean

$\lambda$
Eigenvalue
The stretch factor along the eigenvector.
$\vec v$
Eigenvector
A nonzero vector A doesn't rotate, only scales.

Explanation

An eigenvector is simply a direction that a matrix doesn't turn, only stretches, and its eigenvalue is how much it stretches. The name comes from the German "eigen", meaning "own": these are the matrix's own special directions. They're good for systems because along those directions the matrix acts like plain multiplication, and usually it both turns and stretches. The equation $A\vec v = \lambda\vec v$ looks abstract, but it's just two sides: $A\vec v$ (what the matrix does to the arrow $\vec v$ ) and $\lambda\vec v$ (the same arrow, only scaled by the number $\lambda$ ). And $\det(A - \lambda I) = 0$ is just the test for which numbers $\lambda$ make that possible. That's all there is to it.

Finding them:

Solve the characteristic equation $\det(A - \lambda I) = 0$ . For $2\times2$ matrices it's $\lambda^2 - (\text{trace})\lambda + \det = 0$ .
For each $\lambda$ , solve $(A - \lambda I)\vec v = \vec 0$ . The rows are dependent, so one equation determines the direction.

Think about why the determinant shows up. $A\vec v = \lambda\vec v$ is the same as $(A - \lambda I)\vec v = \vec 0$ . We want a nonzero $\vec v$ that solves this, and a square matrix sends a nonzero vector to $\vec 0$ only when it's not invertible, which means its determinant is $0$ . That's also why the rows come out dependent in step 2: we chose $\lambda$ to make them dependent.

Checks: the eigenvalues add up to the trace and multiply to the determinant.

A common mistake. It's tempting to think the eigenvector for $\lambda$ is a single specific vector, because you write down one answer like $\binom{1}{-1}$ . It isn't: any nonzero multiple, like $\binom{2}{-2}$ or $\binom{-1}{1}$ , works just as well, since scaling $\vec v$ scales both sides of $A\vec v = \lambda\vec v$ . What's fixed is the direction. That's why questions ask you to scale it, like "first component 1".

Why systems care. If $A\vec v = \lambda\vec v$ , then $\vec x = e^{\lambda t}\vec v$ solves $\vec x' = A\vec x$ : the solution moves straight along $\vec v$ , growing or shrinking at rate $\lambda$ . This plays the role the characteristic equation played for second-order equations, and you'll use it in every lesson that follows.

See it

The matrix A = [[2, 1], [1, 2]] acting on a unit vector v as it turns all the way around. Most of the time Av points in a different direction from v. But when v lies along y = x, Av lands on the same line and is 3 times as long, and along y = −x it stays exactly the same. Those two lines are the eigenvectors, with eigenvalues 3 and 1.

Worked examples

Example 1

Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 2 & 7\\ -1 & -6\end{pmatrix}$ .

1
The trace is $-4$ and the determinant is $-5$ , so $\lambda^2 + 4\lambda - 5 = 0$ , which gives $\lambda = 1, -5$ .

Practice

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

Problem 1+5 XP

Find the eigenvalues of $\begin{pmatrix} 2 & 0\\ 0 & 3\end{pmatrix}$ .

Problem 2+5 XP

Find the eigenvalues of $\begin{pmatrix} 2 & 7\\ -1 & -6\end{pmatrix}$ .

Problem 3+5 XP

Give the eigenvector for $\lambda = -5$ , scaled so its first component is 1.

Problem 4+5 XP

What are the eigenvalues of $\begin{pmatrix} 0 & 1\\ -4 & 0\end{pmatrix}$ ?

Problem 5+5 XP

Find the eigenvalues of $\begin{pmatrix} 4 & 1\\ 2 & 3\end{pmatrix}$ .

Problem 6+5 XP

In a two-compartment drug model the amounts obey $\vec x\,' = \begin{pmatrix} -3 & 1\\ 2 & -2\end{pmatrix}\vec x$ . Find the eigenvalues, which are the decay rates of its two modes.

Challenge problem+25 XP

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

For which values of $a$ does $A = \begin{pmatrix} a & 1\\ -4 & 1\end{pmatrix}$ have a repeated eigenvalue? List all of them.

Where it's used

Eigenvalues decide stability everywhere: whether a bridge's vibrations die out, whether an epidemic grows, and which direction PageRank settles into.

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