Definition. Orthogonal gauge orbit [ftip-00AZ]
Definition. Orthogonal gauge orbit [ftip-00AZ]
The orthogonal gauge orbit of \((U,V)\) is
\[ \mathcal O(U,V)=\{(UQ,VQ):Q\in \mathrm O(k)\}. \]Every pair in the orbit represents the same \(W\). This orbit is generally smaller than the complete fibre of the map \((U,V)\mapsto UV^{\mathsf T}\). The restriction to \(\mathrm O(k)\) is geometric: it preserves the Euclidean metric on factor space.