Lattice(matrix, vector) returns the integer lattice generated by matrix and vector. The lattice generated by matrix and vector is the set of all integer points obtained by multiplying matrix on the right by an integer vector and adding vector to that product.

The number of rows of matrix must match the number of entries of vector. Otherwise, an error is raised.

In some literature, a lattice of the d-dimensional Euclidean space consists of all linear combinations with integer coefficients of a basis of that space. In some other literature, see the reference below, the notion of a lattice precisely matches that of this command.

By default, matrix and vector are copied after creating the lattice.

$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$

$\mathrm{with}\left(\mathrm{ZPolyhedralSets}\right)\:$

$\mathrm{with}\left(\mathrm{plots}\right)\:$

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$L≔\mathrm{Lattice}\left(\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,1\right]\right]\right)\,\mathrm{Vector}\left(\left[0\,0\right]\right)\right)$

$L≔\mathrm{Lattice}\left(\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{c}0\\ 0\end{array}\right]\right)$

$\mathrm{plots}:-\mathrm{pointplot}\left(\left[\mathrm{seq}\left(\mathrm{seq}\left(\left[x\,y\right]\,x=-5..5\right)\,y=-5..5\right)\right]\right)$

$A≔\mathrm{Matrix}\left(\left[\left[2\,0\right]\,\left[0\,-1\right]\right]\right)\;$$V≔\mathrm{Vector}\left(\left[0\,0\right]\right)\;$$L≔\mathrm{Lattice}\left(A\,V\right)$

$A≔\left[\begin{array}{cc}2& 0\\ 0& \mathrm{-1}\end{array}\right]$

$V≔\left[\begin{array}{c}0\\ 0\end{array}\right]$

$L≔\mathrm{Lattice}\left(\left[\begin{array}{cc}2& 0\\ 0& \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}0\\ 0\end{array}\right]\right)$

$\mathrm{GetMatrix}\left(L\right)\;$$\mathrm{GetVector}\left(L\right)$

$\left[\begin{array}{cc}2& 0\\ 0& \mathrm{-1}\end{array}\right]$

$\left[\begin{array}{c}0\\ 0\end{array}\right]$

$L≔\left[\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{VectorAdd}\left(\mathrm{MatrixVectorMultiply}\left(A\,\mathrm{Vector}\left(\left[x\,y\right]\right)\right)\,V\right)\,x=-5..5\right)\,y=-5..5\right)\right]$

$L≔\left[\left[\begin{array}{c}\mathrm{-10}\\ 5\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ 5\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ 5\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ 5\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ 5\end{array}\right]\,\left[\begin{array}{c}0\\ 5\end{array}\right]\,\left[\begin{array}{c}2\\ 5\end{array}\right]\,\left[\begin{array}{c}4\\ 5\end{array}\right]\,\left[\begin{array}{c}6\\ 5\end{array}\right]\,\left[\begin{array}{c}8\\ 5\end{array}\right]\,\left[\begin{array}{c}10\\ 5\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ 4\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ 4\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ 4\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ 4\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ 4\end{array}\right]\,\left[\begin{array}{c}0\\ 4\end{array}\right]\,\left[\begin{array}{c}2\\ 4\end{array}\right]\,\left[\begin{array}{c}4\\ 4\end{array}\right]\,\left[\begin{array}{c}6\\ 4\end{array}\right]\,\left[\begin{array}{c}8\\ 4\end{array}\right]\,\left[\begin{array}{c}10\\ 4\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ 3\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ 3\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ 3\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ 3\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ 3\end{array}\right]\,\left[\begin{array}{c}0\\ 3\end{array}\right]\,\left[\begin{array}{c}2\\ 3\end{array}\right]\,\left[\begin{array}{c}4\\ 3\end{array}\right]\,\left[\begin{array}{c}6\\ 3\end{array}\right]\,\left[\begin{array}{c}8\\ 3\end{array}\right]\,\left[\begin{array}{c}10\\ 3\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ 2\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ 2\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ 2\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ 2\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ 2\end{array}\right]\,\left[\begin{array}{c}0\\ 2\end{array}\right]\,\left[\begin{array}{c}2\\ 2\end{array}\right]\,\left[\begin{array}{c}4\\ 2\end{array}\right]\,\left[\begin{array}{c}6\\ 2\end{array}\right]\,\left[\begin{array}{c}8\\ 2\end{array}\right]\,\left[\begin{array}{c}10\\ 2\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ 1\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ 1\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ 1\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ 1\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ 1\end{array}\right]\,\left[\begin{array}{c}0\\ 1\end{array}\right]\,\left[\begin{array}{c}2\\ 1\end{array}\right]\,\left[\begin{array}{c}4\\ 1\end{array}\right]\,\left[\begin{array}{c}6\\ 1\end{array}\right]\,\left[\begin{array}{c}8\\ 1\end{array}\right]\,\left[\begin{array}{c}10\\ 1\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ 0\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ 0\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ 0\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ 0\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ 0\end{array}\right]\,\left[\begin{array}{c}0\\ 0\end{array}\right]\,\left[\begin{array}{c}2\\ 0\end{array}\right]\,\left[\begin{array}{c}4\\ 0\end{array}\right]\,\left[\begin{array}{c}6\\ 0\end{array}\right]\,\left[\begin{array}{c}8\\ 0\end{array}\right]\,\left[\begin{array}{c}10\\ 0\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}0\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}2\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}4\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}6\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}8\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}10\\ \mathrm{-1}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}0\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}2\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}4\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}6\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}8\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}10\\ \mathrm{-2}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}0\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}2\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}4\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}6\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}8\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}10\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}0\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}2\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}4\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}6\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}8\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}10\\ \mathrm{-4}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-10}\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-8}\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-6}\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-4}\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}\mathrm{-2}\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}0\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}2\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}4\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}6\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}8\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}10\\ \mathrm{-5}\end{array}\right]\right]$

$\mathrm{plots}:-\mathrm{pointplot}\left(L\,\mathrm{color}=\mathrm{blue}\right)$

$L≔\mathrm{Lattice}\left(\mathrm{Matrix}\left(\left[\left[1\,0\,0\right]\,\left[0\,1\,0\right]\,\left[0\,0\,1\right]\right]\right)\,\mathrm{Vector}\left(\left[0\,0\,0\right]\right)\right)$

$L≔\mathrm{Lattice}\left(\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\,\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right)$

$\mathrm{plots}:-\mathrm{pointplot3d}\left(\left[\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{seq}\left(\left[x\,y\,z\right]\,x=-5..5\right)\,y=-5..5\right)\,z=-5..5\right)\right]\right)$

Rachid Seghir, Vincent Loechner, and Benoı̂t Meister. "Integer affine transformations of parametric Z-polytopes and applications to loop nest optimization." Proceedings of TACO, Vol. 9(2):8:1–8:27, 2012.

The PolyhedralSets:-ZPolyhedralSets:-Lattice package was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.