invariant subspace

Let $T:V\to V$ 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 $v_1,\mathrm{\dots },v_n$ 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

where $A$ is an $m\times m$ matrix representing the restriction transformation ${T|}_U:U\to U$ relative to the basis $v_1,\mathrm{\dots },v_m$.

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