Orthogonalizable linear representation

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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

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Facts