# Generalized eigenvector

﻿
Generalized eigenvector

In linear algebra, a generalized eigenvector of a matrix "A" is a nonzero vector v, which has associated with it an eigenvalue &lambda; having algebraic multiplicity "k" &ge;1, satisfying

: $\left(A-lambda I\right)^kmathbf\left\{v\right\} = mathbf\left\{0\right\}.$

Ordinary eigenvectors are obtained for "k"=1.

For defective matrices

Generalized eigenvectors are needed to form a complete basis of a defective matrix, which is a matrix in which there are fewer linearly independent eigenvectors than eigenvalues. The generalized eigenvectors "do" form a complete basis, as follows from the Jordan form of a matrix.

In particular, suppose that an eigenvalue &lambda; of a matrix "A" has a multiplicity "m" but only a single corresponding eigenvector $x_1$. We form a sequence of "m" generalized eigenvectors $x_1, x_2, ldots, x_m$ that satisfy:

:$\left(A - lambda I\right) x_k = x_\left\{k-1\right\} !$

for $k=1,ldots,m$, where we define $x_0 = 0$. It follows that:

:$\left(A - lambda I\right)^k x_k = 0. !$

The generalized eigenvectors are linearly independent, but are not determined uniquely by the above relations.

Other meanings of the term

* The usage of generalized eigenfunction differs from this; it is part of the theory of rigged Hilbert spaces, so that for a linear operator on a function space this may be something different.

* One can also use the term "generalized eigenvector" for an eigenvector of the "generalized eigenvalue problem"

: $Av = lambda B v.$

* defective matrix
* eigenvector
* Jordan form

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Eigenvalue, eigenvector and eigenspace — In mathematics, given a linear transformation, an Audio|De eigenvector.ogg|eigenvector of that linear transformation is a nonzero vector which, when that transformation is applied to it, changes in length, but not direction. For each eigenvector… …   Wikipedia

• Eigendecomposition of a matrix — In the mathematical discipline of linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and… …   Wikipedia

• Jordan normal form — In linear algebra, a Jordan normal form (often called Jordan canonical form)[1] of a linear operator on a finite dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some… …   Wikipedia

• Defective matrix — In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly… …   Wikipedia

• List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

• Jordan matrix — In the mathematical discipline of matrix theory, a Jordan block over a ring R (whose identities are the zero 0 and one 1) is a matrix which is composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element… …   Wikipedia

• Karl-Dirk Kammeyer — (* 10. Oktober 1944 in Ahlhorn) ist ein deutscher Forscher im Bereich der Nachrichtentechnik und der Digitalen Signalverarbeitung. Er ist gegenwärtig Professor am Institut für Telekommunikation und Hochfrequenztechnik im Arbeitsbereich… …   Deutsch Wikipedia

• Eigenvalues and eigenvectors — For more specific information regarding the eigenvalues and eigenvectors of matrices, see Eigendecomposition of a matrix. In this shear mapping the red arrow changes direction but the blue arrow does not. Therefore the blue arrow is an… …   Wikipedia

• Eigenvalue algorithm — In linear algebra, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Contents 1 Characteristic polynomial 2 Power… …   Wikipedia

• Principal component analysis — PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by… …   Wikipedia