Orthogonalizable linear representation

Symbol-free definition
A linear representation of a group over a field is termed orthogonalizable if it saisfies the following equivalent conditions:


 * 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

Facts

 * 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.