Eigenvalues and Eigenvectors

An eigenvector of $A$ is a nonzero vector whose direction survives the transformation: $A\mathbf{v} = \lambda\mathbf{v}$. The scalar $\lambda$ is its eigenvalue, the stretch factor along that direction. Everything else rotates or shears; eigenvectors just scale.

Along eigenvector directions, a matrix acts like simple multiplication. Decompose a problem into eigen-directions and hard matrix dynamics become independent scalar dynamics. This is the trick behind PCA, Markov chain steady states, PageRank, and stability analysis.