= Unimodular matrix
{title2=$U\in\mathrm{GL}_n(\mathbb Z)$}
{wiki}
A unimodular matrix is a square <integer matrix> whose <determinant> is $1$ or $-1$. The decisive property is that its <inverse matrix> also has integer entries: multiplication by $U$ gives a <bijection> of the <integer lattice> $\mathbb Z^n$ onto itself. Thus an <invertible linear map> can preserve not only real coordinates but the requirement that every coordinate is an <integer>. These <matrices> form the <integral general linear group>.<Intro to Lattice Algs & Crypto, Mastermath Spring 2018/Lecture 1: Introduction>[§7, Definition 11 and Lemma 12, printed p. 9: determinant and integer-inverse characterization]{ref}
For an original two-dimensional example, take
$$
U=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad
U^{-1}=\begin{pmatrix}1&-1\\0&1\end{pmatrix}.
$$
Its <determinant> is one, and it maps $(a,b)$ to $(a+b,b)$. Conversely, the integer point $(x,y)$ comes from $(x-y,y)$. For instance, $(3,2)$ has new coordinates $(1,2)$ relative to the columns $(1,0)$ and $(1,1)$. No integer point disappears, although the basis vectors and their fundamental cell change shape.
\Image[/unimodular-shear-lattice-bases.png]
{title=Two <lattice bases> generate exactly the same integer points}
{description=Original drawings of the standard basis and the columns of $U=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$. Both displayed cells have area one. The point $(3,2)$ has coordinates $(3,2)$ in the standard basis and $(1,2)$ in the sheared basis.}
This is useful whenever coordinates must remain integral. A <unimodular change of lattice basis> can give a shorter or more convenient description without changing the objects represented. <Smith normal form> and <Hermite normal form> use controlled integer operations to simplify <integer matrices>; ordinary <Gaussian elimination> may divide by numbers that destroy integrality. The geometric freedom is substantial: a sheared basis may look completely different while describing exactly the same infinite <set> of integer points.
A <determinant> of $-1$ is allowed too. Swapping two basis columns reverses <orientation> while preserving the <integer lattice>. The condition is therefore preservation of the complete integer structure, rather than a requirement to preserve orientation.
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