Statistical applications for equivariant matrices (Q5932803)

In a variety of statistical application involving solutions of linear equations, the coefficient matrix is equivariant with respect to a finite group of permutations. It is pointed out how this equivariance property can be used to reduce the cost of computation for solving linear systems. It is shown that the quadratic form is invariant with respect to a permutation matrix. This fact is used to determine the multiplicity of eigenvalues of a matrix and yields the corresponding eigenvectors with low computational cost. Some applications in statistics are presented. These include Fourier transforms on a symmetric group arising in statistical analysis of rankings in an election, and spectral analysis in stationary processes.