Subalgebras of Matrix Algebras Generated by Companion Matrices

Natalio H. Guersenzvaig; Fernando Szechtman

paper

Subalgebras of Matrix Algebras Generated by Companion Matrices

arXiv:1008.1286

Abstract

Let $f, g \in Z [X]$ be monic polynomials of degree $n$ and let $C, D \in M_{n} (Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z < C, D >$ to be a sublattice of finite index in the full integral lattice $M_{n} (Z)$ , in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$ . If $R$ is a commutative ring with identity we determine when $R < C, D >= M_{n} (R)$ , in which case a presentation for $M_{n} (R)$ in terms of $C$ and $D$ is given.