 
Summary: The EigenDecomposition:
Eigenvalues and Eigenvectors
Hervé Abdi1
1 Overview
Eigenvectors and eigenvalues are numbers and vectors associated
to square matrices, and together they provide the eigendecompo
sition of a matrix which analyzes the structure of this matrix. Even
though the eigendecomposition 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 crossproduct matrices. The eigendecomposition 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 eigendecomposition 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.
