Let V be the covariance matrix of the counts C. Regard V = || Vij || as a linear operator in the natural way. Write π(i) for the population index of sample i(1 ≤ i ≤ K). We can write V = Vij as D + W where D is a diagonal matrix with the diagonal element Dii = dπ (i) and Wij = qπ (i),π(j). So the covariance structure depends only on the population labels of the samples. It follows that the vector space of M long column vectors has an orthogonal decomposition into subspaces invariant under V consisting of: 1) a subspace F of vectors whose coordinates are constant within a population. F has dimension K; 2) subspaces Si (1 ≤ i ≤ K). Vectors of Si are zero on samples not belonging to population i, and have coordinate sum 0, which implies that they are orthogonal to F. It now follows that V has K eigenvectors in F, and for each k (1 ≤ i ≤ K), (M(k) − 1 eigenvectors in Sk each of which have
