Solving linear system of equations Ax=b enters into many
scientific applications. In this paper, we consider a special kind
of linear systems, the matrix A 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 A 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
autoregressive processes.
