Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to n-dimensional the value of the time series at a given time). A first step would be to consider the Gramian of the time series $u \in \mathbb R^{n \times \mathbb R}$: $M(u) = u.u' \in \mathbb R^{n \times n}$ (where $'$ is the transpose operator and $.$ is the integration along time). The good point is that $M(u)$ is semi-definite positive and that there is a natural Riemannian distance on the space of positive definite matrices (see the first pages of this paper for a short intro): $$d(M,N) = \|Log(M^{-1}.N) \|_{\mbox{Froebenius}}$$ This is an idea to measure the similarity between the two time series.

For some (convoluted) reason, I am now interested in the matrices of the form $P(u) = u.A.u' \in \mathbb R^{n \times n}$, where $A$ can be seen as a skew-symmetric matrix of $\mathbb R^{\mathbb R \times \mathbb R}$. So I have two questions:

- Is there a natural distance on the set of skew symmetric (or skew hermitian) matrices? I guess the answer is "not in such a general framework". Thus my second question is:
- In the more particular case presented above: $P(u)$ is a Gram-like skew symmetric matrix, do we have such a distance? Note that $A$ is the same for all the time series we want to compare.