I came across the projection kernel, pretty straightforward Kernel defined on pairs of subspaces (represented by the orthonormal matrices P and Q):
$K_p(P, Q) = \|P^TQ\|_F^2$
In another context, I saw this similarity measure over the same space (r is essentially a constant):
$Sim(P, Q) = \frac{\|P^TQ\|_F}{\sqrt{r}} = \sqrt{r^{-1}K_p(P, Q)}$
Is this still a kernel? I am not sure how to think about this, since the square root has a positive and negative solution, but I think in practice we are never interested in the negative solution in this case.