The orthogonal group is the group of all matrices that preserve the dot product
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The orthogonal group is the group of all matrices that preserve the dot product by 
Ciro Santilli 34 Updated 2024-10-13 Created 1970-01-01
Ciro Santilli 34 Updated 2024-10-13 Created 1970-01-01When 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.
This is perhaps the best "default definition".