### 2.46.12 Hermite normal form : `ihermite`

`ihermite` takes as argument a matrix `A` with coefficients
in ℤ.

`ihermite` returns two matrices `U` and `B` such that
`B=U*A`, `U` is invertible in ℤ (det(*U*) = ± 1)
and `B` is upper-triangular. Moreover,
the absolute value of the coefficients above the diagonal of `B` are
smaller than the pivot of the column divided by 2.

The answer is obtained by a Gauss-like reduction algorithm
using only operations of rows with integer coefficients
and invertible in ℤ.

Input :

`A:=[[9,-36,30],[-36,192,-180],[30,-180,180]]; U,B:=ihermite(A)`

Output :

`[[9,-36,30],[-36,192,-180],[30,-180,180]], [[13,9,7],[6,4,3],[20,15,12]],[[3,0,30],[0,12,0],[0,0,60]]`

**Application: Compute a ****ℤ****-basis of the kernel of a
matrix having integer coefficients**

Let `M` be a matrix with integer coefficients.
Input :

`(U,A):=ihermite(transpose(M))`.

This returns *U* and *A* such that `A=U*transpose(M)` hence

`transpose(A)=M*transpose(U)`.

The columns of `transpose(A)` which are identically 0 (at the right,
coming from the rows of *A* which are identically 0 at the bottom)
correspond to columns of `transpose(U)` which form a basis
of `Ker(M)`. In other words, the rows of `A`
which are identically 0 correspond to rows of `U`
which form a basis of `Ker(M)`.

**Example**

Let `M:=[[1,4,7],[2,5,8],[3,6,9]]`. Input

`U,A:=ihermite(tran(M))`

Output

`U:=[[-3,1,0],[4,-1,0],[-1,2,-1]] and A:=[[1,-1,-3],[0,3,6],[0,0,0]]`

Since `A[2]=[0,0,0]`, a ℤ-basis of `Ker(M)` is
`U[2]=[-1,2,-1]`.

Verification `M*U[2]=[0,0,0]`.