Hermite normal form

In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in Rn, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only. Other applications of the Hermite normal form include integer programming, cryptography, and abstract algebra. 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: H is upper triangular (that is, hij = 0 for i > j), and any rows of zeros are located below any other row. The leading coefficient (the first nonzero entry from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it; moreover, it is positive. The elements below pivots are zero and elements above pivots are nonnegative and strictly smaller than the pivot. 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: H is lower triangular, hij = 0 for i < j, and any columns of zeros are located on the right. The leading coefficient (the first nonzero entry from the top, also called the pivot) of a nonzero column is always strictly below of the leading coefficient of the column before it; moreover, it is positive.