# invariant subspace

Let $T:V\rightarrow V$ be a linear transformation of a vector space $V$. A subspace $U\subset V$ is called a $T$ 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 InvariantSubspace 2013-03-22 12:19:55 2013-03-22 12:19:55 rmilson (146) rmilson (146) 9 rmilson (146) Definition msc 15-00 LinearTransformation Invariant