Let \(\mathbf{X}\) be a sample \(\left\{X_{1}, \ldots, X_{n}ight\}\), and let \(A=\left(a_{i j}ight)\) be the matrix of pairwise Euclidean distances \(a_{i j}=\left|X_{i}-X_{j}ight|\). Suppose that \(\pi\) is a permutation of the integers \(\{1, \ldots, n\}\). In a permutation test we re-index one of the samples under a random permutation \(\pi\). For computing \(\mathcal{V}_{n}^{2}\), etc., it is equivalent to re-index both the rows and columns of \(A\) by \(\pi\). Are the reindexing operation and the double-centering operation interchangeable? That is, do we get the same square matrix under (1) and (2)?

(1) Re-index \(A\) then double-center the result.
(2) Double-center \(A\) then re-index the result.
First compare (1) and (2) on a toy example empirically, to see what to prove or disprove. In \(\mathrm{R}\) code, if \(\mathrm{p}=\operatorname{sample}(1: n)\) is the permutation vector, then \(A[p, p]\) is the re-indexed distance matrix.