a normal form for matrices over Z[x]/(x^2-1) ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:57:02Z http://mathoverflow.net/feeds/question/111064 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111064/a-normal-form-for-matrices-over-zx-x2-1 a normal form for matrices over Z[x]/(x^2-1) ? Dima Pasechnik 2012-10-30T09:52:06Z 2012-10-30T09:52:06Z <p>We are discussing, offline, modules over the $\mathbb{Z}$-group ring of the cyclic group of order 2, which is probably better known as the quotient ring $R=\mathbb{Z}[t]/(t^2-1)$. Is there any way to describe matrices over it, in a way similar to Smith Normal Form (SNF), or Hermite Normal Form (HNF)? That is, for $A\in R^{n\times m}$, find $X\in GL(n,R)$ and $Y\in GL(m,R)$, such that $XAY$ is "nice", e.g. diagonal (resp. upper-triangular), like one would get if SNF (resp. HNF) was possible for $R$.</p> <p>I am aware of a similar <a href="http://mathoverflow.net/questions/51942/analogue-of-smith-normal-form-for-matrices-over-mathbb-zt" rel="nofollow">question</a> for $\mathbb{Z}[t]$, which looks harder. One immediate observation is that $A=B+tC$, for $B,C\in \mathbb{Z}^{n\times m}$, and so one can choose $X$, $Y$ to have integer entries, so that $XAY=B'+tC'$, where $C'$ is the SNF of $B'$. </p>