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Encyclopedia > Invariant subspace

In mathematics, an invariant subspace of a linear mapping $T : V rightarrow V$ over some vector space V is a subspace W of V such that T(W) is contained in W. An invariant subspace of T is said to be T invariant. The restriction of T to the invariant subspace is denoted Euclid, detail from The School of Athens by Raphael. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Screenshot (from SSCX Star Warzone). ...

T | W

Certainly V itself, and the subspace {0}, are trivially invariant subspaces for every linear operator VV. There may be no non-trivial invariant subspace of V, such as a rotation of a two-dimensional real vector space. Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...

Another example: let v be an eigenvector of T, i.e. Tv = λv. Then W = span {v} is T invariant. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

## Over finite dimensional vector spaces GA_googleFillSlot("encyclopedia_square");

Over a finite dimensional vector space every linear transformation $T : V rightarrow V$ can be represented via a matrix. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... For the square matrix section, see square matrix. ...

Suppose now W = span { v1, ..., vk} is a T invariant subspace. We shall complete vj into a basis B of V. Then the matrix of T with respect to the basis B will be as follows: In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

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.

## The general case

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 (2005). In case V is only assumed to be a Banach space, it was shown in 1984 by Charles Read that there are counterexamples. In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = âˆ’1. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Charles Read is a British mathematician and currently a professor of mathematics at the University of Leeds. ... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

More generally, invariant subspaces are defined for sets of operators (operator algebras, group representations) as subspaces invariant for each operator in the set. In 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. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

Results from FactBites:

 2 An introduction to nonlinear optimization problem structure (417 words) However, sparsity is not the most important phenomenon associated with a nonlinear function; that role is played by invariant subspaces. The importance of invariant subspaces is that nonlinear information is not required for a function in this subspace. We are particularly interested in functions which have large (as a percentage of the overall number of variables) invariant subspaces.
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