# Generalized Eigenvector

Generalized Eigenvector

In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue λ is greater than its geometric multiplicity (the nullity of the matrix, or the dimension of its nullspace). In such cases, a generalized eigenvector of A is a nonzero vector v, which is associated with λ having algebraic multiplicity k ≥1, satisfying

The set of all generalized eigenvectors for a given λ, together with the zero vector, form the generalized eigenspace for λ.

Ordinary eigenvectors and eigenspaces are obtained for k=1.

Read more about Generalized Eigenvector:  For Defective Matrices, Other Meanings of The Term

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Generalized Eigenvector - Motivation of The Procedure - Ordinary Linear Difference Equations
... A case with a1 = 0 can be excluded, since it represents an equation of less degree ... They have a characteristic polynomial p(x) = xm − amxm−1 − am−1xm−2 −.. ...
Eigenvalue Algorithm - Eigenvalues and Eigenvectors
... or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k ... When k = 1, the vector is called simply an eigenvector, and the pair is called an eigenpair ... Any eigenvalue λ of A has ordinary eigenvectors associated to it, for if k is the smallest integer such that (A - λI)k v = 0 for a generalized eigenvector v, then (A - λI)k-1 v is an ordinary ...

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