I'm reading through this CVPR paper. There's this definition I don't quite understand
Definition 3 (Relative permutation). We define a permutation matrix to be relative if it is the ratio (or difference) of two group elements $(i\to j)$ : $P_{ij} = P_i P_j^T$
I'm not really sure what they mean, but isn't then every permutation matrix a relative one? because if $P_i$ is any matrix then we have $P_i = P_i I^T$, where $I$ is the identity and the identity is a permutation matrix.
Can anyone clarify?
In the definition, only one matrix appears, $P$. In your equation, $P$ and $I$ appear, i.e. two different matrices.
That is, the definition does not say that a permutation matrix is relative if there exists some permutation matrix $Q$ such that $P_{ij} = P_iQ_j^T$