Anyone know anything about these? I was particularly interested, because (1/N) times the matrix of all ones acts like a kind of "zero" in that it wipes out all information about any circulant markov matrix that it multiplies. If Z = matrix([1,1,...,1],[1,1,...,1],...,[1,1,...,1]), and if M is any circulant markov matrix (sum of rows & cols = 1), then Z.M = M.Z = Z Note that |Z.M|=|Z||M|=|M||Z|=|M.Z|=|Z|=0, so this makes sense.