Eigenvectors and eigenvalues #
This file defines eigenspaces, eigenvalues, and eigenvalues, as well as their generalized counterparts. We follow Axler's approach [axler2015] because it allows us to derive many properties without choosing a basis and without using matrices.
An eigenspace of a linear map
f for a scalar
μ is the kernel of the map
(f - μ • id). The
nonzero elements of an eigenspace are eigenvectors
x. They have the property
f x = μ • x. If
there are eigenvectors for a scalar
μ, the scalar
μ is called an eigenvalue.
There is no consensus in the literature whether
0 is an eigenvector. Our definition of
HasEigenvector permits only nonzero vectors. For an eigenvector
x that may also be
x ∈ f.eigenspace μ.
A generalized eigenspace of a linear map
f for a natural number
k and a scalar
μ is the kernel
of the map
(f - μ • id) ^ k. The nonzero elements of a generalized eigenspace are generalized
x. If there are generalized eigenvectors for a natural number
k and a scalar
μ is called a generalized eigenvalue.
The fact that the eigenvalues are the roots of the minimal polynomial is proved in
The existence of eigenvalues over an algebraically closed field
(and the fact that the generalized eigenspaces then span) is deferred to
- [Sheldon Axler, Linear Algebra Done Right][axler2015]
eigenspace, eigenvector, eigenvalue, eigen
The eigenspaces of a linear operator form an independent family of subspaces of
V. That is,
any eigenspace has trivial intersection with the span of all the other eigenspaces.
Eigenvectors corresponding to distinct eigenvalues of a linear operator are linearly independent. (Lemma 5.10 of [axler2015])
We use the eigenvalues as indexing set to ensure that there is only one eigenvector for each
eigenvalue in the image of
The generalized eigenspace for a linear map
f, a scalar
μ, and an exponent
k ∈ ℕ is the
(f - μ • id) ^ k. (Def 8.10 of [axler2015]). Furthermore, a generalized eigenspace for
k is contained in the generalized eigenspace for exponents larger than
If there exists a natural number
k such that the kernel of
(f - μ • id) ^ k is the
maximal generalized eigenspace, then this value is the least such
k. If not, this value is not
For an endomorphism of a Noetherian module, the maximal eigenspace is always of the form kernel
(f - μ • id) ^ k for some
A generalized eigenvalue for some exponent
k is also
a generalized eigenvalue for exponents larger than
The eigenspace is a subspace of the generalized eigenspace.
All eigenvalues are generalized eigenvalues.
Generalized eigenvalues are actually just eigenvalues.
Every generalized eigenvector is a generalized eigenvector for exponent
finrank K V.
(Lemma 8.11 of [axler2015])
Generalized eigenspaces for exponents at least
finrank K V are equal to each other.
f maps a subspace
p into itself, then the generalized eigenspace of the restriction
p is the part of the generalized eigenspace of
f that lies in
p is an invariant submodule of an endomorphism
f, then the
μ-eigenspace of the
p is a submodule of the
Generalized eigenrange and generalized eigenspace for exponent
finrank K V are disjoint.
If an invariant subspace
p of an endomorphism
f is disjoint from the
then the restriction of
p has trivial
The generalized eigenspace of an eigenvalue has positive dimension for positive exponents.
A linear map maps a generalized eigenrange into itself.