International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 1, Pages 53-61
Statistical applications for equivariant matrices
Department of Statistics, King Saud University, P.O. Box 2459, Riyadh 11451, Saudi Arabia
Received 24 November 1999
Copyright © 2001 S. H. Alkarni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Solving linear system of equations enters into many
scientific applications. In this paper, we consider a special kind
of linear systems, the matrix is an equivariant matrix with
respect to a finite group of permutations. Examples of this kind
are special Toeplitz matrices, circulant matrices, and others. The
equivariance property of may be used to reduce the cost of
computation for solving linear systems. We will show that the
quadratic form is invariant with respect to a permutation matrix.
This helps to know the multiplicity of eigenvalues of a matrix and
yields corresponding eigenvectors at a low computational cost.
Applications for such systems from the area of statistics will be
presented. These include Fourier transforms on a symmetric group as
part of statistical analysis of rankings in an election, spectral
analysis in stationary processes, prediction of stationary
processes and Yule-Walker equations and parameter estimation for