eigenvalueeigenvector

An eigenvalue and its corresponding eigenvector of a square matrix A are a scalar λ and a nonzero vector v that satisfy the equation Av = λv. In this relation, the vector v is stretched or compressed by A by a factor of λ and directions are preserved. The equation can be rewritten as (A − λI)v = 0, so nontrivial solutions exist only when det(A − λI) = 0. The polynomial det(A − λI) is called the characteristic polynomial, and its roots are the eigenvalues of A. The eigenvectors associated with each eigenvalue are the nonzero solutions to (A − λI)v = 0.

Eigenvalues can be repeated; their multiplicities are categorized as algebraic multiplicity (multiplicity as a root of the characteristic polynomial) and geometric multiplicity (dimension of the nullspace of A − λI). For a matrix to be diagonalizable, there must be a full set of linearly independent eigenvectors, and in that case A can be written as PDP^{-1}, where D is a diagonal matrix containing the eigenvalues and P has the corresponding eigenvectors as columns.

Computation typically proceeds by first finding the eigenvalues from det(A − λI) = 0, then solving (A − λI)v = 0 for each eigenvalue to obtain the eigenvectors. Real matrices may have complex eigenvalues and eigenvectors.

Applications span many areas. In linear dynamical systems, eigenvalues determine stability and long-term behavior. In data analysis, the eigenvectors of a covariance matrix define principal components. In Markov chains, the eigenvector associated with eigenvalue 1 gives steady-state distributions. In graphics and physics, eigenvectors reveal principal directions of linear transformations.