Various authors may prefer to talk about Hermite normal form in either row-style or column-style. They are essentially the same up to transposition.

An **m** by **n** matrix **A** with integer entries has a (row) Hermite normal form **H** if there is a square unimodular matrix **U** where **H=UA** and **H** has the following restrictions:^{}^{}^{}

The third condition is not standard among authors, for example some sources force non-pivots to be nonpositive^{}^{} or place no sign restriction on them.^{} However, these definitions are equivalent by using a different unimodular matrix **U**. A unimodular matrix is a square invertible integer matrix whose determinant is 1 or -1.

A m by n matrix **A** with integer entries has a (column) Hermite normal form **H** if there is a square unimodular matrix **U** where **H=AU** and **H** has the following restrictions:^{}^{}

Note that the row-style definition has a unimodular matrix **U** multiplying **A** on the left (meaning **U** is acting on the rows of **A**), while the column-style definition has the unimodular matrix action on the columns of **A**. The two definitions of Hermite normal forms are simply transposes of each other.

This page was last edited on 29 May 2018, at 14:13 (UTC).

Reference: https://en.wikipedia.org/wiki/Hermite_normal_form under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Hermite_normal_form under CC BY-SA license.

- Linear Algebra
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