Hermite normal form

In linear algebra, the Hermite normal form is a special form of reduced echelon form over the integers $mathbb\{Z\}$. Specifically, a matrix $M$ over $mathbb\{Z\}$ is said to be in Hermite normal form (often abbreviated HNF) if

* It is upper triangular
* All of its entries are non-negative

If $M$ is a square matrix, we further require that the HNF for $M$ have no zeros along the diagonal. If $M$ is square and $det\; M\; eq\; 0$, then $M\; =\; AU$ where $A$ is in HNF and $U\; in\; GL\_n(mathbb\{Z\})$. Furthermore, $A$, and therefore $U$ are unique.