orthogonal group
Let be a non-degenerate symmetric bilinear form over the real vector space . A linear transformation is said to preserve if for all vectors . The subgroup of the general linear group consisting of all linear transformations that preserve is called the orthogonal group with respect to , and denoted .
If is also positive definite (i.e., is an inner product), then is equivalent to the group of invertible linear transformations that preserve the standard inner product on , and in this case the group is usually denoted .
Elements of are called orthogonal transformations. One can show that a linear transformation is an orthogonal transformation if and only if (i.e., the inverse of equals the transpose of ).
Title | orthogonal group |
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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 |