The existence of an invariant subspace also has a matrix formulation.
Certain linear operators have no proper non-trivial invariant subspace: for instance, rotation of a two-dimensional real vector space.
However, the axis of a rotation in three dimensions is always an invariant subspace.
In fact, the scalar α does not depend on v. The equation above formulates an eigenvalue problem.
Any eigenvector for T spans a 1-dimensional invariant subspace, and vice-versa.
As a consequence of the fundamental theorem of algebra, every linear operator on a nonzero finite-dimensional complex vector space has an eigenvector.
Therefore, every such linear operator in at least two dimensions has a proper non-trivial invariant subspace.
Determining whether a given subspace W is invariant under T is ostensibly a problem of geometric nature.
Matrix representation allows one to phrase this problem algebraically.
Write V as the direct sum W ⊕ W′; a suitable W′ can always be chosen by extending a basis of W. The associated projection operator P onto W has matrix representation A straightforward calculation shows that W is T-invariant if and only if PTP = TP.
Colloquially, a projection that commutes with T "diagonalizes" T. As the above examples indicate, the invariant subspaces of a given linear transformation T shed light on the structure of T. When V is a finite-dimensional vector space over an algebraically closed field, linear transformations acting on V are characterized (up to similarity) by the Jordan canonical form, which decomposes V into invariant subspaces of T. Many fundamental questions regarding T can be translated to questions about invariant subspaces of T. The set of T-invariant subspaces of V is sometimes called the invariant-subspace lattice of T and written Lat(T).
As the name suggests, it is a (modular) lattice, with meets and joins given by (respectively) set intersection and linear span.
In the study of infinite-dimensional operators, Lat(T) is sometimes restricted to only the closed invariant subspaces.
Given a collection T of operators, a subspace is called T-invariant if it is invariant under each T ∈ T. As in the single-operator case, the invariant-subspace lattice of T, written Lat(T), is the set of all T-invariant subspaces, and bears the same meet and join operations.
Given a representation of a group G on a vector space V, we have a linear transformation T(g) : V → V for every element g of G. If a subspace W of V is invariant with respect to all these transformations, then it is a subrepresentation and the group G acts on W in a natural way.
As another example, let T ∈ End(V) and Σ be the algebra generated by {1, T }, where 1 is the identity operator.
Just as the fundamental theorem of algebra ensures that every linear transformation acting on a finite-dimensional complex vector space has a non-trivial invariant subspace, the fundamental theorem of noncommutative algebra asserts that Lat(Σ) contains non-trivial elements for certain Σ. Theorem (Burnside) — Assume V is a complex vector space of finite dimension.
One consequence is that every commuting family in L(V) can be simultaneously upper-triangularized.
If A is an algebra, one can define a left regular representation Φ on A: Φ(a)b = ab is a homomorphism from A to L(A), the algebra of linear transformations on A The invariant subspaces of Φ are precisely the left ideals of A.
A left ideal M of A gives a subrepresentation of A on M. If M is a left ideal of A then the left regular representation Φ on M now descends to a representation Φ' on the quotient vector space A/M.
If [b] denotes an equivalence class in A/M, Φ'(a)[b] = [ab].
The kernel of the representation Φ' is the set {a ∈ A | ab ∈ M for all b}.
The representation Φ' is irreducible if and only if M is a maximal left ideal, since a subspace V ⊂ A/M is an invariant under {Φ'(a) | a ∈ A} if and only if its preimage under the quotient map, V + M, is a left ideal in A.
The invariant subspace problem concerns the case where V is a separable Hilbert space over the complex numbers, of dimension > 1, and T is a bounded operator.
The problem is to decide whether every such T has a non-trivial, closed, invariant subspace.
In the more general case where V is assumed to be a Banach space, Per Enflo (1976) found an example of an operator without an invariant subspace.
A concrete example of an operator without an invariant subspace was produced in 1985 by Charles Read.
Related to invariant subspaces are so-called almost-invariant-halfspaces (AIHS's).
Clearly, every finite-dimensional and finite-codimensional subspace is almost-invariant under every operator.
However, some partial results have been established: for instance, any self-adjoint operator on an infinite-dimensional real Hilbert space admits an AIHS, as does any strictly singular (or compact) operator acting on a real infinite-dimensional reflexive space.