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 :
Output :
Application: Compute a ℤ-basis of the kernel of a
matrix having integer coefficients
Let M be a matrix with integer coefficients.
Input :
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
Output
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].