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Point lattices play an important role in many branches of mathematics, e.g. integer programming, factoring polynomials with rational coefficients, integer factoring and diophantine approximation, to mention some of them.
Let be an ordered set of
independent (column) vectors
in an
-dimensional vector space
a field
. The elements of a set
of vectors which may be written as linear combination of the elements of
![]() | (1) |
with integral coefficients are called lattice vectors, and
is called a (vector) lattice, or just lattice. The vectors
form the lattice basis of
.
Let be the real vector space
.
A typical basis is the so called standard basis given by column vectors with 1 in the
th position,
, and zero otherwise.
The linear independence of the basis vectors ensures the uniqueness of the coefficients in (1).
A basis of a lattice in not determined uniquely. If
is another basis of
, that is the vectors
span
, then there are
such that
![]() | (2) |
Theorem 1. The components of the lattice vectors are obtained by the matrix multiplication (2).
A matrix
is called an integral unimodular matrix if all its elements are integral and its determinant
.
Theorem 2. The lattice bases and
generate the same lattice if and only if one basis goes over to the other by means of an integral unimodular transformation.
There follows that if
are two lattice bases of the same lattice , then the determinants of the coefficient matrices
and
can differ only in sign. Hence the absolute value of these determinants depends only on the lattice
itself. It is called the lattice constant and it is denoted by
, that is
.
The lattice constant has an interesting geometric meaning. To explain this associate with the vector space the Euclidean point space
for which the standard basis
serves as an orthogonal system. Then to each point
and each vector
we can associate a new point
from
. If one chooses a point
as origin, then
together with the basis
form a Cartesian coordinate system
and we can map an arbitrary point
to the unique coordinate
-tuple
through the formula
. Thus the space
can be identified with
.
On the other hand, if is a basis of the vector lattice
this construction allows to assign to every lattice vector lattice a point lattice, namely
Moreover, we can assign to this basis the so called fundamental parallelepiped (or fundamental parallelogram if )
Because
the coordinates of a point
of the fundamental parallelepiped are given by
in
. Using the fact that the standard basis
spans the unit
-dimensional cube of volume one, the volume of the fundamental parallelepiped is
Theorem 3. The volumes of the fundamental parallelepipeds of a lattice are independent of the chosen basis and are equal to the lattice constants.
The fundamental parallelepiped contains no further lattice points in its interior or boundary. Conversely, any set of lattice points with this property determines a lattice basis, and furthermore, it generates the same lattice.
Cite this web-page as:
Štefan Porubský: Point Lattices.