Eigenvectors

An eigenvector of a square matrix A is a nonzero vector v that, when transformed by A, changes by only a scalar factor: Av = λv, where λ is called the eigenvalue associated with v. Equivalently, v lies in a direction that is preserved by the linear transformation represented by A, up to scaling.

To find eigenvalues, one solves the characteristic equation det(A − λI) = 0, where I is the identity matrix. The roots λ of this polynomial are the eigenvalues. For each eigenvalue λ, the corresponding eigenvectors are the nonzero solutions to (A − λI)x = 0; these solutions form the eigenvectors and constitute the eigenspace associated with λ.

Key properties include that eigenvectors corresponding to distinct eigenvalues are linearly independent, and the eigenvectors associated with a repeated eigenvalue form a subspace whose dimension is the geometric multiplicity (which is at most the algebraic multiplicity). A matrix is diagonalizable if it has a full set of n linearly independent eigenvectors, in which case it can be written as A = PDP−1 with D diagonal and P consisting of eigenvectors as columns.

Special cases and examples: real symmetric matrices have real eigenvalues and orthogonal eigenvectors. In general, eigenvectors can be complex. The zero vector is not an eigenvector. Eigenvectors are fundamental in the spectral decomposition of a matrix and appear in numerous applications across science and engineering.

Applications include simplifying linear transformations, principal component analysis, solving systems of differential equations, stability analysis, graph algorithms, and the study of Markov chains.