Hermite normal form

### 15.2.13 Hermite normal form

The Hermite normal form of a matrix A with integer coefficients is a sort of integer row-echelon form. It is an upper triangular matrix B such that B=UA for a matrix U which is invertible in ℤ (det(U)=± 1). The ihermite command finds the Hermite normal form of a matrix.

- ihermite takes A, a matrix with coefficients in ℤ.
- ihermite(A) return a list [U,B] as above, and the absolute value of the elements above the diagonal of B are less than the pivot of the column divided by 2.

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

#### Example

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

⎡ ⎢ ⎢ ⎣ |
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. To find the nullspace of M, you want find what you can multiply M by on the right to get the zero vector, but the Hermite command returns a matrix that you multiply on the left of M. So consider the transpose of M, MT.

Let A be the Hermite normal form of MT, and U an invertible matrix in ℤ such that A=U MT. Transposing this, you will get AT=M UT. Note that M times a column of UT equals the corresponding column of AT. So if a column of AT is the zero vector, then M times the corresponding column of UT will be the zero vector. In fact, these columns of UT will be a ℤ-basis for the nullspace of M.

Any columns of AT which are all zeros correspond to the rows of A which are all zeros. Since A is in Hermite form, these will be at the bottom, and so the corresponding rows of U will be at the bottom, and will be a ℤ-basis for the nullspace of M.

As an example, consider the matrix M:

M:=[[1,4,7],[2,5,8],[3,6,9]]

Find the Hermite decomposition:

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

⎡ ⎢ ⎢ ⎣ |
−3 | 1 | 0
4 | −1 | 0
−1 | 2 | −1
| ⎤ ⎥ ⎥ ⎦
| , |
⎡ ⎢ ⎢ ⎣ |
1 | −1 | −3
0 | 3 | 6
0 | 0 | 0
| ⎤ ⎥ ⎥ ⎦

Only the third row of A consists of all zeros, so a ℤ-basis for the nullspace of M consists of only the third row of U, namely [−1,2,−1].

You can check that this is in the nullspace:

M*U[2]

| ⎡ ⎣ | 0,0,0 | ⎤ ⎦