# Orthogonalizable linear representation

From Groupprops

*This article describes a property to be evaluated for a linear representation of a group, i.e. a homomorphism from the group to the general linear group of a vector space over a field*

*This article provides a semi-basic definition in the following area:* linear representation theory

## Definition

### 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

### Definition with symbols

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## 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.