Invariant subspace

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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.

The existence of invariant subspaces makes it easier to study T.

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.

Another example: let v be an eigenvector of T, i.e. Tv = λv. Then W = span {v} is T invariant. Extending this example, we can show that every linear operator on a complex finite-dimensional vector space with dimension at least 2 has a non-trivial invariant subspace: the eigenvalues of T are the zeros of the characteristic polynomial of T, and this polynomial always has a zero by the fundamental theorem of algebra; we can then pick an eigenvector corresponding to the eigenvalue. This proof doesn't work over the real numbers because not every real polynomial has a real root.

1 Matrix representation
2 Invariant vectors
3 Invariant subspace problem
4 Generalization

[edit] Matrix representation

Over a finite dimensional vector space every linear transformation T : V → V can be represented by a matrix once a basis of V has been chosen. Suppose now W is a T invariant subspace. Pick a basis C = {v₁, ..., v_k} of W and complete it to a basis B of V. Then the matrix [T]_B of T with respect to the basis B will have the following form:

$[T]_B = \begin{bmatrix} [T|_W]_C & * \\ 0 & * \end{bmatrix}$

where the upper-left block express the fact that each image of a vector in W is in W itself and therefore a linear combination of the basis vectors of W.

[edit] Invariant vectors

By simple linear algebra, 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.

[edit] Invariant subspace problem

Main article: Invariant subspace problem

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. It asks whether T always has a non-trivial closed invariant subspace. This problem is unsolved as of 2006. In case V is only assumed to be a Banach space, it was shown in 1975 by P. Enflo and in 1984 by Charles Read that there are counterexamples.

[edit] Generalization

More generally, invariant subspaces are defined for sets of operators (operator algebras, group representations) as subspaces invariant for each operator in the set.

For instance, 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.