Eigenvalues and Eigenvectors

An eigenvector of a square matrix $A$ is a non-zero vector $v$ that the matrix only scales:

The scaling factor $\lambda$ is the eigenvalue. Eigenvectors are special directions where the matrix's action is just stretching by a factor.

Properties:

Trace equals the sum of eigenvalues.Determinant equals the product of eigenvalues.Symmetric real matrices always have real eigenvalues and orthogonal eigenvectors — extremely useful.

Eigenvalues drive the long-run behavior of linear systems. PCA uses eigenvectors of the covariance matrix as principal components. PageRank is an eigenvector. Markov chain stationary distributions are eigenvectors with eigenvalue $1$. The largest-magnitude eigenvalue often dominates long-run dynamics — the spectral gap measures how fast the system mixes.