When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:
This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.

$O(n)=O∈M(n)∣∀x,y,x_{T}y=(Ox)_{T}(Oy)$

This is perhaps the best "default definition".

- All indefinite orthogonal groups of matrices of equal metric signature are isomorphic
- Definition of the indefinite orthogonal group
- Definition of the orthogonal group
- The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
- The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns
- What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms