Eigenvalues and Eigenvectors

The Directions a Matrix Cannot Bend

Apply a matrix to most vectors and they rotate and stretch. But special eigenvectors only get scaled along their line. Their scaling factor is the eigenvalue $\lambda$:

$$Av = \lambda v, \qquad v \ne 0$$

Finding them: Rewrite as $(A - \lambda I)v = 0$. A nonzero solution exists only if $A - \lambda I$ is singular, meaning its determinant is zero:

$$\det(A - \lambda I) = 0$$

This characteristic polynomial yields the eigenvalues as its roots. For each root, solving $(A-\lambda I)v = 0$ gives its eigenspace.

For a triangular matrix, eigenvalues sit right on the diagonal. Always use quick checksums to catch errors before moving forward.

The Eigen-Recipe & Dynamics

Why dynamics care: Repeated multiplication aligns states with the dominant eigenvector (largest $|\lambda|$).

$$|\lambda| > 1: \text{growth} \quad |\lambda| < 1: \text{decay} \quad \lambda = 1: \text{steady}$$

Markov matrices have $\lambda = 1$; their eigenvector gives the long-run market share or Google's PageRank.