# Continuous linear representation of compact group over reals has invariant symmetric positive-definite bilinear form

From Groupprops

## Statement

### For finite-dimensional vector space

Suppose is a compact group and (with finite-dimensional) is a linear representation of over a vector space over , the field of real numbers. Then, there exists a symmetric positive-definite bilinear form such that

In other words, we can think of as a dot product invariant under the action of .

Further, if we choose a basis for that is an orthonormal basis for , then in this basis, all the matrices for , are orthogonal matrices. Thus, another formulation is that every continuous finite-dimensional linear representation over the real numbers is an orthogonalizable linear representation.

## Related facts

- Continuous linear representation of compact group over complex numbers has invariant Hermitian inner product
- Linear representation of finite group over reals has invariant symmetric positive-definite bilinear form
- Linear representation of finite group over complex numbers has invariant Hermitian inner product