  In this paper, we study pure state entanglement in systems of dimension
$2\otimes m\otimes n$. Two states are considered equivalent if they can be
reversibly converted from one to the other with a nonzero probability using
only local quantum resources and classical communication (SLOCC). We introduce
a connection between entanglement manipulations in these systems and the
well-studied theory of matrix pencils. All previous attempts to study general
SLOCC equivalence in such systems have relied on somewhat contrived techniques
which fail to reveal the elegant structure of the problem that can be seen from
the matrix pencil approach. Based on this method, we report the first
polynomial-time algorithm for deciding when two $2\otimes m\otimes n$ states
are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC
equivalence classes in $2\otimes 3\otimes n$ systems, we also determine the
hierarchy between these classes.
