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This paper concerns matrix decompositions in which the factors are restrictedto lie in a closed subvariety of a matrix group. Such decompositions are ofrelevance in control theory: given a target matrix in the group, can it bedecomposed as a product of elements in the subvarieties, in a given order? Andif so, what can be said about the solution set to this problem? Can anirreducible curve of target matrices be lifted to an irreducible curve offactorisations? We show that under certain conditions, for a sufficiently longand complicated such sequence, the solution set is always irreducible, and weshow that every connected matrix group has a sequence of one-parametersubgroups that satisfies these conditions, where the sequence has length lessthan 1.5 times the dimension of the group.

The irreducible control property in matrix groups