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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.


Source: Abdi, Hervé - School of Behavioral and Brain Sciences, University of Texas at Dallas


Collections: Biology and Medicine; Computer Technologies and Information Sciences