Orthogonalizable linear representation
This article provides a semi-basic definition in the following area: linear representation theory
- It is equivalent, as a linear representation, to a linear representation whose image lies completely within the orthogonal group (i.e. it is equivalent to an orthogonal representation)
- There is a nondegenerate, symmetric, bilinear form that is equivalent to the diagonal form, and which is invariant under the action of every element of the group representation
Definition with symbolsPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
- Linear representation of finite group over reals has invariant dot product: It turns out that over the reals, every linear representation of a finite group is orthogonalizable. The proof again relies on an averaging technique. Further, the orthogonalizability can also be used to prove Maschke's averaging lemma for the case of real representations.