invariant subspace

Let $T:V\to V$ be a linear transformation of a vector space $V$. A subspace $U\subset V$ iscalled 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 abasis of $V$ with the first $m$ vectors giving a basis of $U$. Then,the representing matrix of the transformation $T$ relative to thisbasis takes the form

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