Generalized eigenvector
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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
Ordinary eigenvectors are obtained for k=1.
[edit] 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 x1. We form a sequence of m generalized eigenvectors 
that satisfy:
for 
, where we define x0 = 0. It follows that:
The generalized eigenvectors are linearly independent, but are not determined uniquely by the above relations. The reason is that, xk can be replaced by αxk + βxk − 1 for any constants α and β, and still satisfy the equations. Thus, it is typical to choose the generalized eigenvectors to be orthonormal (which is always possible by a Gram–Schmidt process).
[edit] 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 = λBv.
