# orthogonal group

Let $Q$ be a non-degenerate symmetric bilinear form^{} over the real vector space ${\mathbb{R}}^{n}$. A linear transformation $T:V\to V$ is said to *preserve* $Q$ if $Q(Tx,Ty)=Q(x,y)$ for all vectors $x,y\in V$. The subgroup^{} of the general linear group^{} $\mathrm{GL}(V)$ consisting of all linear transformations that preserve $Q$ is called the *orthogonal group ^{}* with respect to $Q$, and denoted $\mathrm{O}(n,Q)$.

If $Q$ is also positive definite^{} (i.e., $Q$ is an inner product), then $\mathrm{O}(n,Q)$ is equivalent to the group of invertible linear transformations that preserve the standard inner product on ${\mathbb{R}}^{n}$, and in this case the group $\mathrm{O}(n,Q)$ is usually denoted $\mathrm{O}(n)$.

Elements of $\mathrm{O}(n)$ are called *orthogonal transformations*.
One can show that a linear transformation $T$ is an orthogonal transformation if and only if ${T}^{-1}={T}^{\mathrm{T}}$ (i.e., the inverse^{} of $T$ equals the transpose^{} of $T$).

Title | orthogonal group |
---|---|

Canonical name | OrthogonalGroup |

Date of creation | 2013-03-22 12:25:54 |

Last modified on | 2013-03-22 12:25:54 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20G20 |

Defines | orthogonal transformation |