Eigenvectors and Linear Transformations

Main idea

Describe a diagonalizable transformation in coordinates built from its own eigenvectors, and all it does is stretch each axis by its eigenvalue. That's what the diagonal matrix $D$ means geometrically.

[T]_{\mathcal B} = P^{-1}AP = D \quad\text{when } \mathcal B \text{ is an eigenvector basis}

Explanation

The matrix of a transformation relative to a basis $\mathcal B$ , written $[T]_{\mathcal B}$ , is simply the matrix that does the same job as $A$ but takes in and hands back $\mathcal B$ -coordinates instead of standard ones. Pick $\mathcal B$ to be eigenvectors and that matrix turns diagonal, so you can see at a glance what the transformation really does. All we're really saying in $[T]_{\mathcal B} = P^{-1}AP = D$ is, read right to left: $P$ (turn eigenvector coordinates into standard ones), $A$ (do the transformation), $P^{-1}$ (turn the answer back into eigenvector coordinates), and the result is $D$ (just stretching each axis by its eigenvalue). It's the same algebra as last lesson, now read as a picture; bear with it.

Why it comes out diagonal. Think about what happens to each basis vector. If $A\vec b_j = \lambda_j\vec b_j$ , then in $\mathcal B$ -coordinates, $\vec b_j$ is just $\vec e_j$ , and it gets sent to $\lambda_j\vec e_j$ . So column $j$ of $[T]_{\mathcal B}$ is $\lambda_j$ in position $j$ and zeros everywhere else. Stack those columns and you get $D$ .

Reading transformations through eigenvalues:

Reflection across a line: eigenvalue $1$ on the line, since those vectors stay put, and $-1$ perpendicular to it, since those get flipped.
Projection onto a line: $1$ on the line, since it's kept, and $0$ perpendicular, since that part gets crushed.
Shear: only eigenvalue $1$ , and only one eigenvector direction (not diagonalizable).
Rotation (not by 0° or 180°): no real eigenvectors at all, since every direction turns.

A common mistake. It's tempting to think every $2\times2$ matrix has at least one real eigenvector, because its characteristic polynomial is a quadratic and you can always run the quadratic formula. It isn't: for rotation by 90°, the characteristic equation is $\lambda^2 + 1 = 0$ , which has no real roots. The roots are complex, and that's the next lesson.

Choosing coordinates that line up with eigenvectors is how complicated-looking matrices are understood. There's an animation further down that shows a matrix turning most vectors but only stretching along two lines.

See it

Here x runs around a circle of radius 2, and Ax = [[1.5, 0.5], [0.5, 1.5]]x follows along, tracing an ellipse. Most of the time Ax points in a different direction from x, but notice what happens on the two eigenlines: along y = x, Ax is exactly twice x, and along y = −x it's just x again. That's why the ellipse reaches out to 4 along y = x but stays at 2 along y = −x. In the basis of those two directions, A is simply the diagonal matrix diag(2, 1).

Worked examples

Example 1

What are the eigenvalues of reflection across the line $y = x$ ?

1
A vector on the line, like $\binom11$ , stays put, so its eigenvalue is 1. A vector perpendicular to the line, like $\binom{1}{-1}$ , gets flipped, so its eigenvalue is $-1$ .

$1$ and $-1$

Practice

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

Problem 1+5 XP

In its eigenvector basis, what is the matrix of $\begin{bmatrix}7 & 2\\ -4 & 1\end{bmatrix}$ (eigenvalue 5 first)? Row by row.

Problem 2+5 XP

What is the only eigenvalue of a shear $\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}$ with $k \ne 0$ ?

Problem 3+5 XP

Does rotation by 90° have real eigenvectors?

Problem 4+5 XP

What are the eigenvalues of a reflection across a line?

Problem 5+5 XP

What are the eigenvalues of a projection onto a line in $\mathbb R^2$ ?

Problem 6+5 XP

A photo filter moves pixel positions by $A = \begin{bmatrix}1.5 & 0.5\\ 0.5 & 1.5\end{bmatrix}$ . Find its matrix relative to the basis $\mathcal B = \left\{\binom11, \binom{1}{-1}\right\}$ , row by row.

Challenge problem+25 XP

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

Let $T$ shift polynomials in $\mathbb P_2$ one unit left: $T(p)(t) = p(t + 1)$ . Find the matrix of $T$ relative to $\{1, t, t^2\}$ , then give the dimension of the eigenspace for its only eigenvalue.

Where it's used

Stress and strain in materials are understood through principal directions (eigenvectors), where a complicated load becomes pure stretching or compression.

3D TransformationsLin Alg · Game Development

Diagonalization Complex Eigenvalues