An application of the Gröbner basis in computation for the minimal polynomials and inverses of block circulant matrices

Abstract

Algorithms for the minimal polynomial and the inverse of a level- n( r 1, r 2,…, r n )-block circulant matrix over any field are presented by means of the algorithm for the Gröbner basis for the ideal of the polynomial ring over the field, and two algorithms for the inverse of a level- n( r 1, r 2,…, r n )-block circulant matrix over a quaternion division algebra are given, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo a prime number.