Eigenvector

An eigenvector of a linear transformation is a non-zero vector that, when the transformation is applied to it, changes only by a scalar factor — the eigenvalue associated with that eigenvector. In matrix terms, if A is a square matrix, v is an eigenvector, and λ is the corresponding eigenvalue, then Av = λv. Eigenvectors are the directions in which a linear map acts by pure scaling, without rotation or shearing, and they form the basis of spectral decomposition — the process of diagonalizing a matrix so that its action becomes transparent.

The significance of eigenvectors extends far beyond linear algebra. In dynamical systems, eigenvectors of the Jacobian matrix at a fixed point determine the stable and unstable manifolds that govern local behavior. In quantum mechanics, eigenvectors of the Hamiltonian operator are the stationary states of a system, and the eigenvalues are the measurable energy levels. In data science, principal component analysis finds the eigenvectors of the covariance matrix, identifying the directions of maximum variance in high-dimensional data.

Not all matrices have a full set of eigenvectors. When they do, the matrix is diagonalizable, and its behavior can be understood by studying the eigenvalues and eigenvectors alone. When they do not, the matrix is defective, and the more general theory of Jordan normal form is required. The question of whether a matrix is diagonalizable is not merely technical — it determines whether the system it represents can be decomposed into independent modes or whether modes are coupled in irreducible ways.