invariant subspace InvariantSubspace 2002-02-15 00:51:19 2007-06-09 15:03:42 Definition \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newtheorem{proposition}{Proposition} Let $T: V\rightarrow V$ be a linear transformation of a vector space $V$. A subspace $U\subset V$ is called a {\em $T$-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 $v_1,\ldots, v_m$.