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 and angles.

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

This is perhaps the best "default definition".

- Definition of the orthogonal group | 8, 391, 4
- Orthogonal group | 0, 1k, 22
- Important Lie group | 0, 5k, 65
- Lie group | 278, 7k, 97
- Differential geometry | 12, 7k, 98
- Geometry | 0, 9k, 154
- Mathematics | 17, 34k, 763
- Ciro Santilli's Homepage | 262, 238k, 4k

- All indefinite orthogonal groups of matrices of equal metric signature are isomorphic | 179
- Definition of the indefinite orthogonal group | 204, 383, 1
- The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose | 102
- The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns | 37
- What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms | 172