 The orthogonal group is the group of all matrices that preserve the dot product

ID: the-orthogonal-group-is-the-group-of-all-matrices-that-preserve-the-dot-product

Ciro Santilli 37  Updated 2025-07-01  +Created 1970-01-01

When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:

O (n) = O \in M (n) ∣\forall x, y, x^{T} y = (O x)^{T} (O y)

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".

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