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Rotations are given by unitary matrices.

What is the geometric meaning of unimodular matrices that are not unitary?

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    $\begingroup$ volume preserving... $\endgroup$
    – Stefan Waldmann
    Commented Sep 27, 2015 at 12:55

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If the matrix has integer elements, then the geometric meaning is that a unimodular transformation 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 $B=CM$.

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    $\begingroup$ @Arul If I understand you correctly, you are wrong. Consider $\left( \begin{smallmatrix} 2 & 0 \\ 0 & 1/2 \end{smallmatrix} \right)$. $\endgroup$
    – David E Speyer
    Commented Sep 27, 2015 at 11:59

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