orthogonal group

Let Q be a non-degenerate symmetric bilinear formMathworldPlanetmath over the real vector space n. A linear transformation T:VV is said to preserve Q if Q(Tx,Ty)=Q(x,y) for all vectors x,yV. The subgroupMathworldPlanetmathPlanetmath of the general linear groupMathworldPlanetmath GL(V) consisting of all linear transformations that preserve Q is called the orthogonal groupMathworldPlanetmath with respect to Q, and denoted O(n,Q).

If Q is also positive definitePlanetmathPlanetmath (i.e., Q is an inner product), then O(n,Q) is equivalent to the group of invertible linear transformations that preserve the standard inner product on n, and in this case the group O(n,Q) is usually denoted O(n).

Elements of O(n) are called orthogonal transformations. One can show that a linear transformation T is an orthogonal transformation if and only if T-1=TT (i.e., the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of T equals the transposeMathworldPlanetmath 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