eigenvector

An eigenvector of a square matrix A is a nonzero vector v such that Av = λv for some scalar λ, called an eigenvalue. In the context of a linear transformation, an eigenvector points in a direction that is stretched or compressed by the transformation by the factor λ, without changing its direction.

To compute eigenvectors, one first finds the eigenvalues by solving det(A − λI) = 0. For each eigenvalue λ, solve (A − λI)v = 0 to obtain the corresponding eigenvectors. The set of all eigenvectors associated with a given λ, together with the zero vector, forms the eigenspace Eλ, a subspace whose dimension is the geometric multiplicity of λ.

Key properties include that eigenvectors associated with distinct eigenvalues are linearly independent. If a matrix has a full set of linearly independent eigenvectors, it is diagonalizable, meaning A can be written as A = PDP⁻¹ with D diagonal and P formed from the eigenvectors. In this case, the action of A is simply to scale each eigenvector by its eigenvalue.

Special cases include the identity matrix, where every nonzero vector is an eigenvector with eigenvalue 1, and the zero matrix, where every vector is an eigenvector with eigenvalue 0.

Applications and methods: eigenvectors reveal invariant directions under a transformation and underpin diagonalization, principal component analysis, vibration analysis, and stability studies. Numerically, power iteration approximates a dominant eigenvector, while the QR algorithm computes a full set of eigenvectors and eigenvalues.