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 λ having algebraic multiplicity "k" ≥1, satisfying

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

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 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 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.

See also

* defective matrix
* eigenvector
* Jordan form

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