invariant subspace

Let T:VV be a linear transformation of a vector spaceMathworldPlanetmath V. A subspacePlanetmathPlanetmathPlanetmath UV is called a T-invariant subspacePlanetmathPlanetmath if T(u)U for all uU.

If U is an invariant subspace, then the restrictionPlanetmathPlanetmath of T to U gives a well defined linear transformation of U. Furthermore, suppose that V is n-dimensional and that v1,,vn is a basis of V with the first m vectors giving a basis of U. Then, the representing matrix of the transformationMathworldPlanetmath T relative to this basis takes the form


where A is an m×m matrix representing the restriction transformation T|U:UU relative to the basis v1,,vm.

Title invariant subspace
Canonical name InvariantSubspace
Date of creation 2013-03-22 12:19:55
Last modified on 2013-03-22 12:19:55
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Definition
Classification msc 15-00
Related topic LinearTransformation
Related topic InvariantMathworldPlanetmath