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In mathematics, an invariant subspace of a linear mapping over some vector space V is a subspace W of V such that

, literally: T(W) is contained in W.

An invariant subspace of T is said to be T invariant. T is also denoted

Certainly V itself, and the subspace {0}, are trivially invariant subspaces, regardless of T. There may be no non-trivial invariant subspace of V, such as 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.

1 Over finite dimensional vector spaces

Over a finite dimensional vector space every linear transformation can be represented via a matrix.

Suppose now W = span { v₁, ... , v_k} is a T invariant subspace. We shall complete v_j into a basis B of V. Then the matrix of T with respect to the basis B will be as follows:

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

2 The general case

The invariant subspace problem concerns the case where V is a 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 (2004). In case V is only assumed to be a Banach spaceFunctional analysis In mathematics, Banach spaces named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. Definition Banach s, it was shown in 1984 by Charles Read that there are counterexampleIn logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. a specific instance of the falsity of a universal quantification (a "for all" statement). For example, consider thes.

More generally, invariant subspaces are defined for sets of operators ( operator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. In particular, it is a set of operas, group representationRepresentation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. Representation theory is important because it enables many group-theoretic problems to be rs) as subspaces invariant for each operator in the set.