Orthogonal Group and the Dot Product
An invertible matrix preserves the norm exactly when it preserves dot products: .
Lemma 9.9 ( And the Dot Product)
Proof
If preserves dot products, then it preserves norms by taking . Conversely, if , the polarisation identity applied to and gives
Thus the two conditions are equivalent.
Related
Stated in
- Lemma 9.9 ( And the Dot Product)§9.3 Orthogonal Groups