An eigenvector of a matrix is a non-zero vector that the matrix does not turn: after the transformation it still points along the same line, only stretched, shrunk or flipped. The eigenvalue is the number it gets multiplied by. In symbols, Av = λv, where A is the matrix, v the eigenvector and λ (lambda) the eigenvalue. For the matrix with rows (2, 1) and (1, 2), the vector (1, 1) becomes (3, 3), three times as long on the same line, so it is an eigenvector with eigenvalue 3. Most vectors are knocked off their line. The eigenvectors are the directions where the matrix acts like simple stretching.

Start with the picture, not the formula

A 2 by 2 matrix is a rule for moving every point in the plane. The first column says where the arrow (1, 0) lands, the second column says where (0, 1) lands, and everything else follows. Our matrix throughout this page is

A = | 2  1 |
    | 1  2 |

which sends (1, 0) to (2, 1) and (0, 1) to (1, 2). Draw a grid of arrows from the origin, apply A to each, and watch what happens.

• (1, 0) goes to (2, 1). It started flat along the x axis and now points up and to the right. Knocked off its line.
• (0, 1) goes to (1, 2). Knocked off its line the other way.
• (1, 1) goes to (2 + 1, 1 + 2) = (3, 3). Same direction, three times longer.
• (1, -1) goes to (2 - 1, 1 - 2) = (1, -1). Exactly where it started. Same direction, length multiplied by 1.

The last two are the eigenvectors. Every arrow on the diagonal line through (1, 1) gets tripled. Every arrow on the line through (1, -1) stays put. Everything else gets pulled towards the (1, 1) direction.

Note that an eigenvector is really a whole line of them. (2, 2) and (-5, -5) are also eigenvectors with eigenvalue 3, because doubling or flipping the arrow does not change its line. That is why answers usually give one representative, like (1, 1).

The eigenvalues mean something on the picture

Our matrix has eigenvalues 3 and 1: it stretches the plane by 3 along one diagonal and leaves the other diagonal alone.

Finding them without guessing

Guessing (1, 1) worked because the matrix is friendly. The general method comes straight from the definition.

Av = λv can be rewritten as (A - λI)v = 0, where I is the identity matrix. We want a non-zero v that the matrix A - λI sends to zero. That only happens when A - λI squashes the plane flat, which means its determinant is zero.

Step 1, the characteristic equation.

A - λI has rows (2 - λ, 1) and (1, 2 - λ). Its determinant is

(2 - λ)(2 - λ) - (1)(1) = λ² - 4λ + 4 - 1 = λ² - 4λ + 3.

Set it to zero: λ² - 4λ + 3 = 0, which factors as (λ - 3)(λ - 1) = 0. So λ = 3 or λ = 1.

Step 2, the eigenvector for λ = 3.

A - 3I has rows (-1, 1) and (1, -1). We need -x + y = 0, so y = x. Any vector (x, x) works; take (1, 1).

Step 3, the eigenvector for λ = 1.

A - I has rows (1, 1) and (1, 1). We need x + y = 0, so y = -x. Take (1, -1).

Step 4, check. A(1, 1) = (2·1 + 1·1, 1·1 + 2·1) = (3, 3). A(1, -1) = (2·1 + 1·(-1), 1·1 + 2·(-1)) = (1, -1). Both match.

Two quick sanity checks that work for any 2 by 2 matrix: the eigenvalues add up to the trace (the diagonal sum, 2 + 2 = 4, and 3 + 1 = 4), and they multiply to the determinant (2·2 - 1·1 = 3, and 3·1 = 3). If your eigenvalues fail either check, there is an arithmetic slip.

Why anyone cares: repeated multiplication

Eigenvectors turn a hard question into an easy one. What happens if you apply A ten times to (1, 0)?

Multiplying matrices ten times is tedious. Instead, write (1, 0) in terms of the two eigenvectors:

(1, 0) = ½(1, 1) + ½(1, -1).

Each time A is applied, the (1, 1) part gets multiplied by 3 and the (1, -1) part by 1. So after n applications:

Aⁿ(1, 0) = ½·3ⁿ(1, 1) + ½(1, -1).

Check n = 2: ½·9(1, 1) + ½(1, -1) = (4.5 + 0.5, 4.5 - 0.5) = (5, 4). Multiplying out directly, A(1, 0) = (2, 1) and A(2, 1) = (5, 4). It matches.

For n = 10, 3¹⁰ = 59,049, so the answer is (29,524.5 + 0.5, 29,524.5 - 0.5) = (29,525, 29,524). The vector is now pointing almost exactly along (1, 1). That is the general lesson: apply a matrix again and again and vectors swing towards the eigenvector with the largest eigenvalue. Population models, Markov chains, Google's original PageRank and the vibration modes of a bridge all rest on this idea.

Matrices with fewer eigenvectors

Not every matrix has two nice real eigenvector lines, and this is a favourite exam trap.

• A 90 degree rotation, rows (0, -1) and (1, 0), turns every arrow. Its characteristic equation is λ² + 1 = 0, which has no real solutions. No real eigenvectors.
• A shear, rows (1, 1) and (0, 1), slides the grid sideways. Only the x axis stays on its line, with eigenvalue 1 repeated. One eigenvector line instead of two.

Both make good final scenes in a video, because the picture shows instantly why the algebra comes out the way it does.

Make it

To make your own version for class, script the picture first and the algebra second. A five-minute video is about 750 words at 150 words a minute, which comfortably fits the sections above. A suggested scene list: the grid and the question; four arrows transformed, two knocked off and two not; the definition Av = λv; the eigenvalue table; the characteristic equation worked line by line; the check; repeated multiplication swinging towards (1, 1); the rotation that has none.

1. 1

   Pick one matrix and keep it all the way through

   Use small whole numbers with whole-number eigenvalues, like rows (2, 1) and (1, 2). Changing example halfway loses the viewer.

2. 2

   Do all the arithmetic on paper first

   Work out the eigenvalues, the eigenvectors and the trace and determinant checks before you write a word of script.

3. 3

   Write one paragraph per scene, picture first

   Describe what the arrows do before you show any symbols. Say each equation in words as well, such as 'lambda squared minus four lambda plus three equals zero'.

4. 4

   Paste the script into openCanviz

   Choose Keep my wording so the maths is said exactly as you wrote it, and choose the whiteboard style so each grid and arrow is drawn as it is explained.

5. 5

   Check every sign and arrow

   Pause on each scene. Arrows must land where the arithmetic says, and every minus sign must be there. Fix any wrong one in the editor scene by scene.

A drawn tool is good at the step-by-step diagrams and spoken algebra. It is not a substitute for a coded animation where the grid smoothly shears in real time; how to make a maths YouTube channel like 3Blue1Brown on a budget explains when that extra effort is worth it, and works a different matrix. For the general approach to a marked maths video, see how to make a math explainer video for a class assignment and how to make a diagram animate step by step.

Common questions

Can an eigenvector be the zero vector? No, by definition. A·0 = λ·0 for every λ, so allowing it would make every number an eigenvalue. Eigenvalues can be zero; eigenvectors cannot.

Are eigenvectors unique? Only up to scaling. Any non-zero multiple of an eigenvector is also an eigenvector with the same eigenvalue, so (1, 1), (3, 3) and (-1, -1) all describe the same line.

Do eigenvectors have to be at right angles? Not in general. They are at right angles for symmetric matrices, like the one on this page, which is why (1, 1) and (1, -1) are perpendicular. For a non-symmetric matrix they usually are not.

What does a negative eigenvalue look like? The arrow stays on its line but flips to point the other way. A reflection across the x axis, rows (1, 0) and (0, -1), has eigenvalue -1 along the y axis: (0, 1) goes to (0, -1).