A generalized eigenvector for an 
matrix 
is a vector 
for which
 |
for some positive integer 
. Here, 
denotes the 
identity matrix.
The smallest such 
is known as the generalized
eigenvector order of the generalized eigenvector. In this case, the value 
is the generalized eigenvalue to which
is associated and the linear span of all generalized
eigenvectors associated to some generalized
eigenvalue 
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.
Indeed, for such matrices, at least one eigenvalue 
has geometric multiplicity larger than its algebraic
multiplicity, thereby implying that the collection
of linearly independent eigenvectors
of 
is "too small" to be a basis of 
. In particular, the aim of determining the generalized eigenvectors
of an 
matrix 
is to "enlarge" the set of linearly
independent eigenvectors of such a matrix
in order to form a basis for 
.
