# Generalized eigenvector

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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

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