# 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

$$\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill 0\hfill & \hfill C\hfill \end{array}\right)$$ |

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^{} |