<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %A matint.xml GAP documentation Alexander Hulpke --> <!-- %A Thomas Breuer --> <!-- %A Rob Wainwright --> <!-- %% --> <!-- %% --> <!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland --> <!-- %Y Copyright (C) 2002 The GAP Group --> <!-- %% -->
<Chapter Label="Integral matrices and lattices">
<Heading>Integral matrices and lattices</Heading>
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<Section Label="Linear equations over the integers and Integral Matrices">
<Heading>Linear equations over the integers and Integral Matrices</Heading>
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<Section Label="Normal Forms over the Integers">
<Heading>Normal Forms over the Integers</Heading>
This section describes the computation of the Hermite and Smith normal form
of integer matrices.
<P/>
The Hermite Normal Form (HNF) <M>H</M> of an integer matrix <M>A</M> is
a row equivalent upper triangular form such that all off-diagonal entries
are reduced modulo the diagonal entry of the column they are in.
There exists a unique unimodular matrix <M>Q</M> such that <M>Q A = H</M>.
<P/>
The Smith Normal Form <M>S</M> of an integer matrix <M>A</M> is
the unique equivalent diagonal form with <M>S_i</M> dividing <M>S_j</M> for
<M>i < j</M>.
There exist unimodular integer matrices <M>P, Q</M> such that <M>P A Q = S</M>.
<P/>
All routines described in this section build on the <Q>workhorse</Q> routine
<Ref Func="NormalFormIntMat"/>.
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<Section Label="Determinant of an integer matrix">
<Heading>Determinant of an integer matrix</Heading>
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