Cheatsheet: Eigenvectors and eigenvalues

The definition

M · v = λ · v       (v nonzero)

An eigenvector v is a direction the transformation only scales (stays on its own line). The eigenvalue λ is the scaling factor. For an eigenvector, the whole matrix acts like a single number.

Finding them

1. Eigenvalues: solve the characteristic equation det(M - λI) = 0.
2. Eigenvectors: for each λ, find the null space of (M - λI) (the vectors it crushes to zero).

Works because M·v = λ·v rearranges to (M - λI)·v = 0, which has a nonzero solution only when M - λI collapses, i.e. its determinant is zero (the inverses-lesson condition).

The diagonal basis (change-of-basis payoff)

With two independent eigenvectors as the columns of P:

D = P^-1 · M · P    is diagonal, eigenvalues on the diagonal

In the eigenvector basis the transformation is pure scaling along the axes, the simplest it can look.

Worked examples

Matrix	Eigenvalues	Eigenvectors
[[2,0],[0,3]] (stretch)	2, 3	[1,0], [0,1] (already diagonal)
[[0,-1],[1,0]] (rotation)	none real	none (everything rotates off its line)
[[1,1],[0,1]] (shear)	1 (only)	[1,0] only (degenerate)
[[3,1],[0,2]]	3, 2	[1,0], [1,-1]

Last one in full: det([[3-λ,1],[0,2-λ]]) = (3-λ)(2-λ) = 0 gives λ = 3, 2. Check: M·[1,0]=[3,0]=3·[1,0]; M·[1,-1]=[2,-2]=2·[1,-1]. With P=[[1,1],[0,-1]], D = P^-1·M·P = [[3,0],[0,2]].

Why it matters for AI (substantive)

• PCA: eigenvectors of the data’s covariance matrix are the directions of greatest variance (principal components); eigenvalues say how much each captures.
• Exploding / vanishing gradients: signal passing repeatedly through a weight matrix is scaled by its eigenvalues; magnitude above 1 explodes, below 1 vanishes.
• Spectral graph neural networks use eigenvectors of a graph’s matrix; PageRank is an eigenvector problem.

Pitfalls to dodge

• Eigenvectors are lines, not single arrows. Any nonzero scalar multiple is the same eigenvector.
• Not every matrix has real eigenvectors. Rotations have none; shears have one.
• Zero eigenvalue is meaningful. It means that direction collapses to the origin (not invertible).
• Eigenvalue vs eigenvector. λ is the number; v is the direction.

The one-line version

Eigenvectors are the directions a transformation only scales, the eigenvalue is the scale factor, and in the eigenvector basis the transformation becomes a diagonal matrix of those factors, the simplest description it has.