# invariant

Let $A$ be a set, and $T:A\to A$ a transformation of that set. We say that $x\in A$ is an invariant of $T$ whenever $x$ is fixed by $T$:

$$T(x)=x.$$ |

We say that a subset $B\subset A$ is invariant with respect to $T$ whenever

$$T(B)\subset B.$$ |

If this is so, the restriction^{} of $T$
is a well-defined transformation of the invariant subset:

$${T|}_{B}:B\to B.$$ |

The definition generalizes readily to a family of transformations with common domain

$${T}_{i}:A\to A,i\in I$$ |

In this case we say that a subset is invariant, if it is invariant with respect to all elements of the family.

Title | invariant |
---|---|

Canonical name | Invariant |

Date of creation | 2013-03-22 12:26:09 |

Last modified on | 2013-03-22 12:26:09 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 8 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 03E20 |

Related topic | Transformation |

Related topic | InvariantSubspace |

Related topic | Fixed |