Eigenvalues and eigenvectors

Most vectors change direction when a matrix multiplies them. A few special ones do not: they are only stretched or shrunk. These are eigenvectors, and the stretch factor is the eigenvalue λ:

A·v = λ·v

A 2×2 example

A = [[2, 1], [1, 2]] sends (1, 1) to (3, 3): the same direction, 3 times longer. So (1, 1) is an eigenvector with eigenvalue 3. It sends (1, −1) to (1, −1): eigenvalue 1.

Finding them exactly

For a 2×2 matrix, the eigenvalues solve the characteristic equation

λ² − (trace)·λ + det = 0

where the trace is a + d, the sum of the diagonal. For our A: λ² − 4λ + 3 = 0, so λ = 3 or 1. If the equation has no real roots, the matrix rotates every direction and has no real eigenvectors.

Power iteration

Multiply any starting vector by A again and again, rescaling each time. The direction with the biggest eigenvalue grows fastest and takes over.

A = [[2, 1],
     [1, 2]]
v = [1.0, 0.0]
for step in range(10):
    w = [A[0][0] * v[0] + A[0][1] * v[1], A[1][0] * v[0] + A[1][1] * v[1]]
    size = max(abs(w[0]), abs(w[1]))
    v = [w[0] / size, w[1] / size]
print("direction ≈", [round(x, 4) for x in v])
print("stretch ≈", round(size, 4))

trace, det = 2 + 2, 2 * 2 - 1 * 1
root = (trace * trace - 4 * det) ** 0.5
print("exact eigenvalues:", (trace + root) / 2, "and", (trace - root) / 2)

After ten steps the vector points along (1, 1) and the stretch factor is 3, matching the exact answer.

Where it is used

• Google's PageRank is the main eigenvector of the web's link matrix, found by power iteration.
• PCA (principal component analysis) finds the directions in which data varies most: the eigenvectors of its covariance matrix.
• Bridges and buildings vibrate at frequencies given by eigenvalues; engineers check them so that wind or footsteps cannot shake a structure apart.

Common mistakes

• Counting the zero vector as an eigenvector. A·0 = λ·0 for every λ, so it tells you nothing; eigenvectors must be non-zero.

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