- Generalized eigenvector
In

linear algebra , a**generalized eigenvector**of a matrix "A" is a nonzero vector**v**, which has associated with it aneigenvalue λ havingalgebraic multiplicity "k" ≥1, satisfying: $(A-lambda\; I)^kmathbf\{v\}\; =\; mathbf\{0\}.$

Ordinary

eigenvector s 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 fewerlinearly independent eigenvectors than eigenvalues. The generalized eigenvectors "do" form a complete basis, as follows from theJordan form of a matrix.In particular, suppose that an eigenvalue λ 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:

:$(A\; -\; lambda\; I)\; x\_k\; =\; x\_\{k-1\}\; !$

for $k=1,ldots,m$, where we define $x\_0\; =\; 0$. It follows that:

:$(A\; -\; lambda\; I)^k\; x\_k\; =\; 0.\; !$

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 ofrigged Hilbert space s, so that for alinear operator on afunction space this may be something different.* One can also use the term "generalized eigenvector" for an eigenvector of the "

generalized eigenvalue problem ": $Av\; =\; lambda\; B\; v.$

**See also***

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