invariant subspace

Let $T:V\rightarrow V$ be a linear transformation of a vector space $V$ . A subspace $U\subset V$ is called a $T$ -invariant subspace if $T(u)\in U$ for all $u\in U$ .

If $U$ is an invariant subspace, then the restriction of $T$ to $U$ gives a well defined linear transformation of $U$ . Furthermore, suppose that $V$ is $n$ -dimensional and that $v_{1},\ldots,v_{n}$ is a basis of $V$ with the first $m$ vectors giving a basis of $U$ . Then, the representing matrix of the transformation $T$ relative to this basis takes the form

\begin{pmatrix}A&B\\ 0&C\end{pmatrix}

where $A$ is an $m\times m$ matrix representing the restriction transformation $T\big{|}_{U}:U\to U$ relative to the basis $v_{1},\ldots,v_{m}$ .

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	Invariant