eigenarvovector

In linear algebra, an eigenarvovector (eigenvector) of a square matrix A is a nonzero vector v for which A v = λ v holds for some scalar λ, called the eigenvalue corresponding to v. The pair (v, λ) is an eigenpair of A. For real matrices, eigenvalues may be real or complex; complex eigenvalues occur in conjugate pairs.

To compute eigenpairs, one solves the characteristic equation det(A − λI) = 0 to find the eigenvalues λ. For each λ, one solves (A − λI) v = 0 to obtain the corresponding eigenvectors v, determined up to a nonzero scaling factor.

Geometric interpretation: an eigenvector v is a direction that is preserved by the linear transformation A, with the transformation acting as a pure scaling by λ along that direction. Other vectors generally change direction and magnitude in more complex ways.

Diagonalization and multiplicities: If A has n linearly independent eigenvectors, it is diagonalizable: A = P D P^{-1}, where D contains the eigenvalues on its diagonal. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of det(A − λI) = 0, while the geometric multiplicity is the dimension of its eigenspace. A matrix is diagonalizable when the geometric multiplicity equals the algebraic multiplicity for all eigenvalues.

Applications and methods: Eigenvectors identify natural scaling directions in data and systems. They underpin principal component analysis, stability analysis, differential equations, and power iteration methods for finding dominant eigenvalues. For large matrices, numerical algorithms such as the QR algorithm or inverse iteration are commonly used. In particular, real symmetric matrices have real eigenvalues and orthogonal eigenvectors, a consequence of the spectral theorem.