|
|
|
Viewing Version
5
of
'invariant subspace'
|
[ view 'invariant subspace'
|
back to history
]
| Title of object: |
invariant subspace |
| Canonical Name: |
InvariantSubspace |
| Type: |
Definition |
| Created on: |
2002-02-15 00:51:19 |
| Modified on: |
2005-08-02 23:11:19 |
| Classification: |
msc:15-00 |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newtheorem{proposition}{Proposition} |
Content:
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 $U$
defines 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 $v_1,\ldots, v_m$. |
|
|
|
|
|
