I would like to construct a set of $k$ orthogonal matrices in $\mathbb{R}^{n \times n}$ with maximal summed pairwise distance (in terms of L2 operator norm). Any ideas? I am thinking of just doing SGD. Is this something that has been studied?
We can restrict ourselves to determinant = 1, if that helps. In that case, it seems like we would just get out the $k$-th roots of unity (up to a phase shift) for $n=2$.