sage: A = matrix(ZZ, 2, [1,2, 3,4]); A [1 2] [3 4] sage: Bstar, mu = A.gramm_schmidt()

The rows of the matrix are obtained from the rows of by the Gramm-Schmidt procedure.

sage: Bstar [ 1 2] [ 4/5 -2/5] sage: mu [ 0 0] [11/5 0]

A *lattice*
is a subgroup that
is free of rank such that
.

For example, the basis
,
for a lattice
is *not*
LLL reduced because and

sage: A = matrix(ZZ, 2, [1,2, 3,4]) sage: A.LLL() [1 0] [0 2]

William Stein 2012-09-24