- Generalized eigenvector
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 independenteigenvectors than eigenvalues. The generalized eigenvectors "do" form a complete basis, as follows from the Jordan formof a matrix.
In particular, suppose that an eigenvalue λ of a matrix "A" has a multiplicity "m" but only a single corresponding eigenvector . We form a sequence of "m" generalized eigenvectors that satisfy:
for , where we define . It follows that:
The generalized eigenvectors are linearly independent, but are not determined uniquely by the above relations.
Other meanings of the term
* The usage of
generalized eigenfunctiondiffers from this; it is part of the theory of rigged Hilbert spaces, so that for a linear operatoron a function spacethis may be something different.
* One can also use the term "generalized eigenvector" for an eigenvector of the "
generalized eigenvalue problem"
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