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