Given a square 
nonsingular integer matrix 
, there exists an 
unimodular matrix 
and an 
matrix 
(known as the Hermite normal form of 
) such that
 |
Specifying a certain set of conditions on the elements of 
makes it (and therefore 
) unique. One possible set of conditions (corresponding to
the "columns" version and making 
lower triangular)
is given by
1. 
for 
,
2. 
for all 
,
and
3. 
and 
for 
(Domich et al. 1987).
For a complexity analysis of Hermite normal form computation, see Storjohann and Labahn (1996).
The Hermite normal form for integer matrices is implemented in the Wolfram Language
as HermiteDecomposition[A],
which however uses the "rows" convention (thus making 
upper triangular)
and replaces condition (3) with balanced remainders (mod 
).
