eigendecomposition

Eigendecomposition, or spectral decomposition, of a square matrix A, is a factorization of the form A = PDP^{-1}, where D is a diagonal matrix whose diagonal entries are eigenvalues λ1, ..., λn and P is invertible with columns being the corresponding eigenvectors v1, ..., vn. Such a decomposition exists precisely when A is diagonalizable, i.e., when there exist n linearly independent eigenvectors. If A has real entries, the eigenvalues may be real or complex; the decomposition is usually considered over the field of interest (real or complex). When eigenvectors are taken as columns, P^{-1}AP = D.

How computed: eigenvalues satisfy det(A − λI) = 0; for each eigenvalue λ, the corresponding eigenvectors solve (A − λI)x = 0. The geometric multiplicity (dimension of the eigenspace) must equal the algebraic multiplicity for each eigenvalue for A to be diagonalizable.

Special case: real symmetric (or Hermitian) matrices are always diagonalizable with a real spectrum, and eigenvectors can be chosen orthonormal. In that case A = QΛQ^T with Q orthogonal and Λ diagonal.

Limitations: not every matrix is diagonalizable; non-defective matrices admit a Jordan form, but diagonalization is not guaranteed. Numerically, eigenvalues can be computed by QR algorithms; large matrices may require iterative methods.

Applications: simplifying repeated powers A^k via A^k = PD^kP^{-1}, solving linear differential equations, principal component analysis, stability and dynamics, and more.