(A notational issue is that in our genetic data, if m is the number of individuals, then the square matrix for which we calculate eigenvalues has rank m − 1 [we lose a dimension by forcing each column to sum to zero]. We will, for the majority of the paper, write m′ = m − 1 for the number of nonzero eigenvalues. However in this theoretical section, we will assume there are m nonzero eigenvalues.)
