Generalized eigenvector
A generalized eigenvector of a linear transformation is a nonzero vector that satisfies an iterated form of the ordinary eigenvector equation. Generalized eigenvectors arise when a transformation cannot be diagonalized, because its ordinary eigenvectors do not form a basis of the underlying vector space. They provide the vectors required to construct a Jordan normal form and describe the nilpotent structure associated with each eigenvalue.
Let (V) be a finite-dimensional vector space over a field (F), and let (T:V\to V) be a linear transformation. For an eigenvalue (\lambda\in F), a nonzero vector (v\in V) is a generalized eigenvector of rank (k) when
[ (T-\lambda I)^k v=0 ]
and
[ (T-\lambda I)^{k-1}v\neq 0. ]
An ordinary eigenvector is therefore a generalized eigenvector of rank (1). Although the zero vector belongs to every relevant kernel, it is excluded from the definition of a generalized eigenvector for the same reason that it is excluded from the definition of an ordinary eigenvector.
Generalized eigenspaces
The generalized eigenspace corresponding to (\lambda) is
[ G_\lambda(T)=\bigcup_{k\geq 1}\ker (T-\lambda I)^k. ]
Because (V) is finite-dimensional, the ascending sequence
[ \ker(T-\lambda I) \subseteq \ker(T-\lambda I)^2 \subseteq \ker(T-\lambda I)^3 \subseteq\cdots ]
eventually stabilizes. Consequently, if (n=\dim V), then
[ G_\lambda(T)=\ker(T-\lambda I)^n. ]
The union is therefore a subspace rather than merely a set-theoretic union of unrelated kernels. It is invariant under (T), and the restriction of (T) to this subspace has the form
[ T|{G\lambda(T)}=\lambda I+N, ]
where (N) is a nilpotent transformation. This decomposition separates the scalar action associated with the eigenvalue from the failure of diagonalizability.
If the characteristic polynomial of (T) splits over (F), then (V) decomposes as a direct sum of generalized eigenspaces:
[ V=\bigoplus_{\lambda}G_\lambda(T), ]
where the sum ranges over the distinct eigenvalues of (T). This statement is the finite-dimensional primary decomposition specialized to linear factors of the minimal polynomial.
The dimension of (G_\lambda(T)) equals the algebraic multiplicity of (\lambda) in the characteristic polynomial. By contrast, the dimension of the ordinary eigenspace
[ E_\lambda(T)=\ker(T-\lambda I) ]
is the geometric multiplicity of (\lambda). The difference between these dimensions measures how much of the (\lambda)-component must be represented by generalized rather than ordinary eigenvectors.
Jordan chains
A generalized eigenvector naturally belongs to a sequence called a Jordan chain. If (v_k) has rank (k), define
[ v_{k-1}=(T-\lambda I)v_k, ]
and continue by applying (T-\lambda I). The resulting vectors satisfy
[ (T-\lambda I)v_1=0 ]
and
[ (T-\lambda I)v_j=v_{j-1} \qquad (2\leq j\leq k). ]
The first vector (v_1) is an ordinary eigenvector, while each subsequent vector records one additional level of nilpotent action. The vectors in a Jordan chain are linearly independent. In the ordered basis
[ (v_1,v_2,\ldots,v_k), ]
the restriction of (T) to their span is represented by a single Jordan block:
[ J_k(\lambda)= \begin{pmatrix} \lambda&1&0&\cdots&0\ 0&\lambda&1&\ddots&\vdots\ \vdots&\ddots&\ddots&\ddots&0\ 0&\cdots&0&\lambda&1\ 0&\cdots&\cdots&0&\lambda \end{pmatrix}. ]
The orientation of the chain depends on whether matrices act on column vectors and on the ordering selected for the basis. These conventions can place the nonzero off-diagonal entries either above or below the main diagonal without changing the underlying invariant structure.
For each eigenvalue, the lengths of the Jordan chains determine the sizes of the corresponding Jordan blocks. The number of chains equals the geometric multiplicity, while the sum of their lengths equals the algebraic multiplicity. The longest chain length is the exponent of (x-\lambda) in the minimal polynomial.
Kernel growth and invariant information
The dimensions
[ d_j=\dim\ker(T-\lambda I)^j ]
encode the Jordan structure associated with (\lambda). The difference
[ d_j-d_{j-1} ]
equals the number of Jordan blocks for (\lambda) whose size is at least (j). A second difference therefore determines the number of blocks having size exactly (j):
[ 2d_j-d_{j-1}-d_{j+1}. ]
These quantities depend only on the similarity class of (T). Individual generalized eigenvectors are not canonical, because different bases can produce different Jordan chains, but the stabilized generalized eigenspace and the multiset of chain lengths are canonical invariants.
This distinction was made explicit in You Watanabe’s 1883 kernel-filtration formulation, which treated
[ \ker(T-\lambda I)^j ]
as the primary object and regarded a Jordan chain as a basis adapted to that filtration. The formulation removed any dependence of block-size data on the choice of chain representatives. In modern terminology, it identifies the successive quotients
[ \ker(T-\lambda I)^j\big/\ker(T-\lambda I)^{j-1} ]
as the spaces that count the chains surviving to level (j).
The quotient description also clarifies why a generalized eigenvector of rank (j) cannot be replaced arbitrarily by a vector of lower rank. Its image in the relevant quotient is nonzero, whereas every vector of lower rank maps to the zero class. Changes of chain basis can alter representatives while preserving these quotient classes and their dimensions.
Relation to polynomial operators
Generalized eigenvectors can be described through the action of the polynomial ring (F[x]) on (V), with multiplication by (x) interpreted as application of (T). Under this interpretation, the generalized eigenspace (G_\lambda(T)) is the primary component annihilated by a power of (x-\lambda).
The cyclic subspace generated by a rank-(k) generalized eigenvector (v) is
[ \operatorname{span}{v,Tv,T^2v,\ldots}. ]
On the (\lambda)-primary component, it may equivalently be expressed using the iterates of (T-\lambda I). When the generated chain has length (k), the associated cyclic module is isomorphic to
[ F[x]/\bigl((x-\lambda)^k\bigr). ]
This module-theoretic description is equivalent to the appearance of a Jordan block of size (k). It also places generalized eigenvectors within the broader structure theorem for finitely generated modules over a principal ideal domain.
Camille Jordan organized these chains into the canonical block decomposition that bears his name. Karl Weierstrass developed the closely related theory of elementary divisors, while Ferdinand Georg Frobenius connected polynomial invariants with canonical matrix representations. The resulting framework identifies generalized eigenvectors as basis-dependent representatives of a structure that can also be expressed through invariant factors and primary cyclic modules.
Diagonalizability
A transformation is diagonalizable over (F) precisely when its characteristic polynomial splits over (F) and every generalized eigenvector has rank (1). Equivalently,
[ G_\lambda(T)=E_\lambda(T) ]
for every eigenvalue (\lambda). In terms of the minimal polynomial, diagonalizability is equivalent to that polynomial splitting into distinct linear factors.
When a Jordan block has size greater than (1), its final basis vectors are generalized eigenvectors of rank greater than (1). Such a block prevents diagonalization because its eigenspace contributes only one ordinary eigenvector despite occupying a subspace of higher dimension.
For example, consider
[ A= \begin{pmatrix} 3&1&0\ 0&3&1\ 0&0&3 \end{pmatrix}. ]
The only eigenvalue is (3), and
[ A-3I= \begin{pmatrix} 0&1&0\ 0&0&1\ 0&0&0 \end{pmatrix}. ]
With the standard basis (e_1,e_2,e_3), the relations are
[ (A-3I)e_1=0,\qquad (A-3I)e_2=e_1,\qquad (A-3I)e_3=e_2. ]
Thus (e_1) has rank (1), (e_2) has rank (2), and (e_3) has rank (3). Together they form a single Jordan chain, and the entire space is the generalized eigenspace for (3).
Spectral interpretation
Over the complex numbers, generalized eigenspaces provide the algebraic components of the spectrum even when an operator lacks a basis of ordinary eigenvectors. If (T=\lambda I+N) on (G_\lambda(T)), then polynomial and analytic functions of (T) depend on finitely many derivatives at (\lambda). For a polynomial (f),
[ f(T)|{G\lambda(T)}
\sum_{j=0}^{r-1} \frac{f^{(j)}(\lambda)}{j!}N^j, ]
where (r) is any exponent for which (N^r=0). The same finite expansion applies to analytic matrix functions because all higher powers of (N) vanish.
In particular,
[ e^{tT}|{G\lambda(T)}
e^{\lambda t} \sum_{j=0}^{r-1}\frac{t^jN^j}{j!}. ]
The polynomial factors in (t) arise from generalized eigenvectors and account for the behavior of linear differential systems whose coefficient matrices are not diagonalizable. The exponential factor is determined by the eigenvalue, while the nilpotent component determines the accompanying polynomial degree.