Hermite normal form

In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers ${\displaystyle \mathbb {Z} }$. Just as reduced echelon form can be used to solve problems about the solution to the linear system ${\displaystyle Ax=b}$ where ${\displaystyle x\in \mathbb {R} ^{n}}$, the Hermite normal form can solve problems about the solution to the linear system ${\displaystyle Ax=b}$ where this time ${\displaystyle x}$ is restricted to have integer coordinates only. Other applications of the Hermite normal form include integer programming, cryptography, and abstract algebra.