Editing Invariant subspace (section)

== For a single operator ==
Consider a vector space <math>V</math> and a linear map <math>T: V \to V.</math>  A subspace <math>W \subseteq V</math> is called an '''invariant subspace for''' <math>T</math>, or equivalently, {{Mvar|T}}-invariant, if {{Mvar|T}} transforms any vector <math>\mathbf{v} \in W</math> back into {{Mvar|W}}.  In formulas, this can be written<math display="block">\mathbf{v} \in W \implies T(\mathbf{v}) \in W</math>or<ref>{{harvnb|Roman|2008|loc=p. 73 §2}}</ref> <math display="block">TW\subseteq W\text{.}</math>

In this case, {{Mvar|T}} [[restriction (mathematics)|restricts]] to an [[endomorphism]] of {{Mvar|W}}:<ref>{{harvnb|Roman|2008|loc=p. 73 §2}}</ref><math display="block">T|_W : W \to W\text{;}\quad T|_W(\mathbf{w}) = T(\mathbf{w})\text{.}</math>

The existence of an invariant subspace also has a [[Matrix representation|matrix formulation]].  Pick a [[basis (linear algebra)|basis]] ''C'' for ''W'' and complete it to a basis ''B'' of ''V''.  With respect to {{Mvar|B}}, the operator {{Mvar|T}} has form <math display=block> T = \begin{bmatrix} T|_W & T_{12} \\ 0 & T_{22} \end{bmatrix} </math> for some {{Math|''T''<sub>12</sub>}} and {{Math|''T''<sub>22</sub>}}, where <math>T|_W</math> here denotes the matrix of <math>T|_W</math> with respect to the basis ''C''.