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| In [[linear algebra]], the '''Hermite normal form''' is an analogue of [[reduced echelon form]] for [[matrix (mathematics)|matrices]] over the [[integer]]s '''Z'''.
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| ==Nonsingular square matrices==
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| A [[nonsingular matrix|nonsingular]] [[square matrix]] ''M'' = (''m''<sub>ij</sub>) with integer entries is in Hermite normal form (HNF) if
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| * ''M'' is [[upper triangular]],<ref>Some authors prefer using [[lower triangular]] matrices; suitable adjustments must be made to the rest of the definition</ref>
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| * its diagonal entries, ''m''<sub>ii</sub>, are positive,
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| * for j < i, ''m''<sub>ii</sub> > ''m''<sub>ji</sub> ≥ 0, i.e. in a given column, the entries above the diagonal are non-negative and less than the diagonal.
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| ==General matrices==
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| More generally, an ''m''×''n'' matrix with integer entries is in (HNF) if there exists
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| * ''r'' with 0 ≤ ''r'' ≤ ''n'',
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| * a [[strictly increasing]] function f: [''r'' + 1, ''n''] → [1, ''m''],
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| such that the first ''r'' columns of ''M'' are zero, and for ''r'' + 1 ≤ ''j'' ≤ ''n''
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| * ''m''<sub>f(j)j</sub> > 0,
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| * ''m''<sub>ij</sub> = 0 when ''i'' > f(''j''),
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| * ''m''<sub>f(j)j</sub> > ''m''<sub>kj</sub> ≥ 0 when ''k'' < ''f(j)''.
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| ==Uniqueness of the Hermite normal form==
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| Given any ''m''×''n'' matrix ''A'' with integer entries, there is a unique ''m''×''n'' matrix ''H'', in HNF, with integer entries such that
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| :<math>H=UA</math> with ''U'' ∈ GL<sub>''n''</sub>('''Z''') (i.e. ''U'' is [[unimodular matrix|unimodular]]). | |
| The matrix formed by the nonzero columns of ''H'' is called the '''Hermite normal form of ''A'''''.
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| ===Example===
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| The HNF of matrix A is matrix H.
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| :<math> | |
| A=\begin{pmatrix}
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| 3 & 3 & 1 & 4 \\
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| 0 & 1 & 0 & 0 \\
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| 0 & 0 & 19 & 16 \\
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| 0 & 0 & 0 & 3
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| \end{pmatrix}
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| \qquad
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| H=\begin{pmatrix}
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| 3 & 0 & 1 & 1\\
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| 0 & 1 & 0 & 0\\
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| 0 & 0 &19 & 1\\
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| 0 & 0 & 0 & 3
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| \end{pmatrix}
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| </math>
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| ===Example===
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| The HNF of matrix A is matrix H.
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| <math>
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| A=
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| \begin{pmatrix}
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| 0&0&5 & 0 & 1 & 4 \\
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| 0&0&0 & -1 & -4 & 99 \\
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| 0&0&0 & 20 & 19 & 16 \\
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| 0&0&0 & 0 & 2 & 1\\
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| 0&0&0 & 0 & 0 & 3\\
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| 0&0&0 & 0 & 0 & 0
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| \end{pmatrix}
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| \qquad H=
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| \begin{pmatrix}
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| 0& 0& 5& 0& 0& 2\\
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| 0& 0& 0& 1& 0& 1\\
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| 0& 0& 0& 0& 1& 2\\
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| 0& 0& 0& 0& 0& 3\\
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| 0& 0& 0& 0& 0& 0\\
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| 0& 0& 0& 0& 0& 0\\
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| \end{pmatrix}
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| </math>
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| Here we have r=2; f(3)=1, f(4)=2, f(5)=3, f(6)=2.
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| (f(j) gives the row of the lowest nonzero entry in column j.)
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| ==See also==
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| *[[Hermite ring]]
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| ==Notes==
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| <references/>
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| ==References==
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| *Section 2.4.2 of {{Citation
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| | last=Cohen
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| | first=Henri
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| | author-link=Henri Cohen (number theorist)
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| | title=A Course in Computational Algebraic Number Theory
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| | publisher=[[Springer-Verlag]]
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| | location=Berlin, New York
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| | series=Graduate Texts in Mathematics
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| | isbn=978-3-540-55640-4
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| | mr=1228206
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| | year=1993
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| | volume=138
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| }}
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| [[Category:Linear algebra]]
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| {{numtheory-stub}}
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