Eigendecomposition

An eigendecomposition (or eigenvalue decomposition) of a square matrix $ is a factorization of the form = Q \Lambda Q^{-1}$, where $\Lambda$ is a diagonal matrix of eigenvalues and $ is a matrix whose columns are the corresponding eigenvectors. It exists if and only if $ is diagonalizable. The eigendecomposition reveals the intrinsic coordinate system in which the linear operator acts as independent scalings along orthogonal axes. It is the computational backbone of principal component analysis, spectral clustering, and the analysis of linear dynamical systems. When an eigendecomposition does not exist, the Jordan normal form provides the closest alternative, though at the cost of losing the simplicity that makes modal analysis tractable.