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The Eigen-Decomposition: Eigenvalues and Eigenvectors

The Eigen-Decomposition: Eigenvalues and EigenvectorsHerv Abdi11 OverviewEigenvectorsandeigenvaluesare numbers and vectors associatedto square matrices, and together they provide theeigen-decompo-sitionof a matrix which analyzes the structure of this matrix. Eventhough 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 thistype of matrices is important in statistics because it is used to findthe maximum (or minimum) of functions involving these matri-ces. For example, principal component analysis is obtained fromthe eigen-decomposition of a covariance matrix and gives the leastsquare estimate of the original data and Eigenvalues are also referred to ascharacter-istic vectors and latent rootsorcharacteristic equation(in German, eigen means specific of or characteristic of ).

0 1 0 0 ‚ does not have eigenvalues. Even when a matrix has eigenvalues and eigenvectors, the computation of the eigenvectors and eigenvalues of a matrix requires a large number of computations and is therefore better performed by com-puters. 2.1 Digression: An infinity of eigenvectors for one eigenvalue

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