Concentrated Matrix Exponential Distributions

We revisit earlier attempts for finding matrix exponential (ME) distributions of a given order with low coefficient of variation (cv). While there is a long standing conjecture that for the first non-trivial order, which is order 3, the cv cannot be less than 0.200902 but the proof of this conjecture is still missing. In previous literature ME distributions with low cv are obtained from special subclasses of ME distributions (for odd and even orders), which are conjectured to contain the ME distribution with minimal cv. The numerical search for the extreme distribution in the special ME subclasses is easier for odd orders and previously computed for orders up to 15. The numerical treatment of the special subclass of the even orders is much harder and extreme distribution had been found only for order 4. In this work, we further extend the numerical optimization for subclasses of odd orders (up to order 47), and also for subclasses of even order (up to order 14). We also determine the parameters of the extreme distributions, and compare the properties of the optimal ME distributions for odd and even order. Finally, based on the numerical results we draw conclusions on both, the behavior of the odd and the even orders.