- 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
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