The Eigen-Decomposition: Eigenvalues and Eigenvectors Summary: The Eigen-Decomposition: Eigenvalues and Eigenvectors Hervé Abdi1 1 Overview Eigenvectors and eigenvalues are numbers and vectors associated to square matrices, and together they provide the eigen-decompo- sition of a matrix which analyzes the structure of this matrix. Even though the eigen-decomposition does not exist for all square ma- trices, it has a particularly simple expression for a class of matri- ces often used in multivariate analysis such as correlation, covari- ance, or cross-product matrices. The eigen-decomposition of this type of matrices is important in statistics because it is used to find the maximum (or minimum) of functions involving these matri- ces. For example, principal component analysis is obtained from the eigen-decomposition of a covariance matrix and gives the least square estimate of the original data matrix. Eigenvectors and eigenvalues are also referred to as character- istic vectors and latent roots or characteristic equation (in German, "eigen" means "specific of" or "characteristic of"). The set of eigen- values of a matrix is also called its spectrum.