A remark on symmetric circulant matrices

Abstract

By using the elementary properties of symmetric circulant matrices, we present another method for obtaining the solution for the linear system of equations Ax = b where A is an n× n nonsingular symmetric real circulant matrix, i.e., we have a formula for obtaining A -1. Let n = p k 1 1 p k 2 2· p k s s be the prime-power d mposition of n, and A= G 1⊗ G 2⊗·⊗ G s , where G i is an p k i i × p k i symmetric real circulant matrix for i=1,2,·, s. Then we show the existence of a permutation matrix T such that TAT -1 is a symmetric real circulant matrix. We also consider the case of A being an mn× mn symmetric block of symmetric real circulant matrices.