  This paper concerns matrix decompositions in which the factors are restricted
to lie in a closed subvariety of a matrix group. Such decompositions are of
relevance in control theory: given a target matrix in the group, can it be
decomposed as a product of elements in the subvarieties, in a given order? And
if so, what can be said about the solution set to this problem? Can an
irreducible curve of target matrices be lifted to an irreducible curve of
factorisations? We show that under certain conditions, for a sufficiently long
and complicated such sequence, the solution set is always irreducible, and we
show that every connected matrix group has a sequence of one-parameter
subgroups that satisfies these conditions, where the sequence has length less
than 1.5 times the dimension of the group.
