Doublecentering

Double-centering is a matrix operation used to transform a square matrix into a centered form by removing its row and column effects and then adding the overall mean. The resulting matrix has zero row sums and zero column sums, which makes it useful for revealing intrinsic structure in the data. It is commonly applied to distance or similarity matrices and to data matrices in multivariate analysis.

Given an n-by-n matrix A, the double-centering procedure computes row means r_i = (1/n) sum_j a_ij, column means c_j = (1/n) sum_i a_ij, and the grand mean m = (1/n^2) sum_i sum_j a_ij. The double-centered entry is a*_ij = a_ij − r_i − c_j + m. If A is symmetric, the row and column means align so that a*_ij = a*_ji and the matrix is centered in both dimensions.

In classical multidimensional scaling (MDS), double-centering is applied to the squared distance matrix D^2 to obtain the Gram matrix B, via B = −1/2 J D^2 J, where J = I − (1/n) 11^T. The Gram matrix B contains inner products and can be decomposed spectrally to recover coordinates in Euclidean space. Double-centering thus links pairwise distances to a centered inner-product representation.

Properties and caveats include that double-centering enforces zero sums across rows and columns; if the original distances arise from an exact Euclidean embedding, the resulting matrix is positive semidefinite. If the distances are not Euclidean, B may have negative eigenvalues, signaling non-Euclidean structure in the data. The operation is also used in kernel methods to produce centered kernel matrices from raw similarity measures.