If the matrix has integer elements, then the geometric meaning is that a <A HREF="http://mathworld.wolfram.com/UnimodularTransformation.html">unimodular transformation</A> maps the integer lattice onto itself:

> Consider a basis $B$ of an $m$-dimensional lattice
> $L(B)=\{Bx:x\in\mathbb{Z}^m$\}, and another basis $C$, then $L(B)=L(C)$
> if and only if there exists a unimodular matrix $M$ (an $m\times m$
> matrix with integer entries and determinant $\pm 1$) such that $B=CM$.