Gruber P. Convex and Discrete Geometry

Relations Between Different Bases
21 Lattices 357
Different bases of a given lattice are related in a rather simple way. Recall, an integer
unimodular d × d matrix is a d × d matrix U with integer entries and det U =±1.
Theorem 21.1. Let {b1,...,bd} be a basis of a lattice L in E d . Then the following
statements hold:
(i) Let {c1,...,cd} be another basis of L. Then
(1)
c1 = u11b1 +···+u1dbd
c2 = u21b1 +···+u2dbd
........................
cd = ud1b1 +···+uddbd
or (c1,...,cd) = (b1,...,bd)U T ,
where U = (uik) is a suitable integer unimodular d × d matrix.
(ii) Let U = (uik) be an integer unimodular d × d matrix and let {c1,...,cd} be
defined by (1). Then {c1,...,cd} is a basis of L.
Proof. (i) Since {b1,...,bd} is a basis of L, each vector of L is an integer linear
combination of b1,...,bd. This implies (1), where U = (uik) is an integer d × d
matrix. Noting that {c1,...,cd} is also a basis of L, it follows that, conversely,
(2)
b1 = v11c1 +···+v1dcd
b2 = v21c1 +···+v2dcd
.......................
bd = vd1c1 +···+vddcd
or (b1,...,bd) = (c1,...,cd)V T ,
where V = (vik) is a suitable integer d × d matrix. From (1) and (2) we conclude
that
(3) (c1,...,cd) = (b1,...,bd)U T = (c1,...,cd)V T U T = (c1,...,cd)(UV) T
Since c1,...,cd are linearly independent and thus (c1,...,cd) a non-singular d × d
matrix, it follows from (3) that det(UV) T = 1 or det U det V = 1. Since U and
V are integer matrices, their determinants are also integers. This then shows that
det U =±1, concluding the proof of (i).
(ii) Since U is an integer unimodular d × d matrix, (1) implies that c1,...,cd
are in L, are linearly independent and
(b1,...,bd) = (c1,...,cd)V T , where V = U −1 .
Being the inverse of the integer unimodular d × d matrix U, the matrix V is also
an integer unimodular matrix. Thus each bi is an integer linear combination of the
vectors c1,...,cd. Since each vector of L is an integer linear combination of the
vectors b1,...,bd, it follows that each vector of L is an integer linear combination
of the vectors c1,...,cd. Since c1,...,cd are linearly independent, {c1,...,cd}
is a basis. ⊓⊔