Pull on a rubber sheet and most arrows drawn on it get bent and turned — but a few special directions only get longer or shorter, never rotated. Those unturning directions are eigenvectors, and how much they stretch is the eigenvalue.
Special directions of a transformation
For a square matrix A, an eigenvector is a nonzero vector v whose direction is unchanged when A acts on it — A only scales it: Av = λv. The scale factor λ is the eigenvalue. These special directions reveal a transformation's fundamental behavior and underpin PCA, vibration analysis, and Google's PageRank.
Along an eigenvector, the matrix acts like simple multiplication by λ. Eigenvalues are found by solving det(A − λI) = 0, the characteristic equation.
For A = [[2, 0], [0, 3]], the vector [1, 0] is an eigenvector with eigenvalue 2, and [0, 1] has eigenvalue 3 — the axes are only stretched.
- Take A = [[2, 0], [0, 3]] and the vector v = [1, 0].
- Compute Av.
- Check whether the result is a scalar multiple of v, and name λ.
What you should see: Av = [2, 0] = 2·[1, 0], so v is an eigenvector with eigenvalue λ = 2.