A generalized eigenvector for an 
matrix 
is a nonzero vector 
for which
 |
for some positive integer 
and scalar 
. Here, 
denotes the 
identity matrix.
The smallest such 
is known as the generalized eigenvector order of the generalized
eigenvector. In this case, 
is necessarily an eigenvalue
of 
to which 
is associated, and the linear span of all generalized
eigenvectors associated to 
is known as the generalized eigenspace for 
.
As the name suggests, generalized eigenvectors are generalizations of eigenvectors of the usual kind; more precisely, an eigenvector
is a generalized eigenvector corresponding to 
.
Generalized eigenvectors are of particular importance for 
matrices 
which fail to be diagonalizable.
Over an algebraically closed field such as 
, this occurs iff at least one eigenvalue 
has geometric multiplicity smaller than its algebraic
multiplicity, thereby implying that the collection
of linearly independent eigenvectors
of 
is "too small" to be a basis. Generalized
eigenvectors can then be used to "enlarge" the set of linearly
independent eigenvectors of such a matrix in order
to form a basis.
