Now we carry out a singular value decomposition on the matrix M. (A standard reference for the numerical methods is [19]. Public domain software is readily available—we used the well-known package LAPACK, http://www.netlib.org/lapack.) We are chiefly interested here in the case that the number of samples is less than the number of markers: m < n. Computationally we will form the sample covariance of the columns of M. The resulting matrix is m × m, with a dimension equal to the number of samples in the dataset. We then compute an eigenvector decomposition of X. Eigenvectors corresponding to “large” eigenvalues are exposing nonrandom population structure. This means that a central issue for this paper is what is “large” here, or, more precisely, what is the distribution of the largest eigenvalues of X at random (when there is no population structure)?
