Invariant Subspace

In mathematics, an invariant subspace of a linear mapping

T : V → V

from some vector space V to itself is a subspace W of V such that T(W) is contained in W. An invariant subspace of T is also said to be T invariant.

If W is T-invariant, we can restrict T to W to arrive at a new linear mapping

T|W : W → W.

Next we give a few immediate examples of invariant subspaces.

Certainly V itself, and the subspace {0}, are trivially invariant subspaces for every linear operator T : V → V. For certain linear operators there is no non-trivial invariant subspace; consider for instance a rotation of a two-dimensional real vector space.

Let v be an eigenvector of T, i.e. T v = λv. Then W = span {v} is T invariant. As a consequence of the fundamental theorem of algebra, every linear operator on a complex finite-dimensional vector space with dimension at least 2 has an eigenvector. Therefore every such linear operator has a non-trivial invariant subspace. The fact that the complex numbers are algebraically closed is required here. Comparing with the previous example, one can see that the invariant subspaces of a linear transformation are dependent upon the underlying scalar field of V.

An invariant vector (fixed point of T), other than 0, spans an invariant subspace of dimension 1. An invariant subspace of dimension 1 will be acted on by T by a scalar, and consists of invariant vectors if and only if that scalar is 1.

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 is 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.

More generally, invariant subspaces are defined for sets of operators as subspaces invariant for each operator in the set. Let L(V) denote the algebra of linear transformations on V, and Lat(T) be the family of subspaces invariant under T ∈ L(V). (The "Lat" notation refers to the fact that Lat(T) forms a lattice; see discussion below.) Give a nonempty set Σ ⊂ L(V), one considers the invariant subspaces invariant under each T ∈ Σ. In symbols,

For instance, it is clear that if Σ = L(V), then Lat(Σ) = { {0}, V}.

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 ∈ L(V) and Σ be the algebra generated by {1, T}, where 1 is the identity operator. Then Lat(T) = Lat(Σ). Because T lies in Σ trivially, Lat(Σ) ⊂ Lat(T). On the other hand, Σ consists of polynomials in 1 and T, therefore the reverse inclusion holds as well.