**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

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**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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