  In this paper, we prove that there is a natural correspondence between
product identities for theta functions and integer matrix exact covering
systems. We show that since $\mathbb{Z}^n$ can be taken as the disjoint union
of a lattice generated by $n$ linearly independent vectors in $\mathbb{Z}^n$
and a finite number of its translates, certain products of theta functions can
be written as linear combinations of other products of theta functions. We
firstly give a general theorem to write a product of $n$ theta functions as a
linear combination of other products of theta functions. Many known identities
for products of theta functions are shown to be special cases of our main
theorem. Several entries in Ramanujan's notebooks as well as new identities are
proved as applications, including theorems for products of three and four theta
functions that have not been obtained by other methods.
