  For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is
a matroid K on the union of S and T that, like the direct sum and the free
product, has the restriction of K to S equal to M and the contraction of K to T
equal to N. We abstract a matrix construction to get a general matroid
construction: the matroid union of any rank-preserving extension of M on the
union of S and T with the direct sum of N and the rank-0 matroid on S is a
semidirect sum of M and N. We study principal sums in depth; these are such
matroid unions where the extension of M has each element of T added either as a
loop or freely on a fixed flat of M. A second construction of semidirect sums,
defined by a Higgs lift, also specializes to principal sums. We also explore
what can be deduced if M and N, or certain of their semidirect sums, are
transversal or fundamental transversal matroids.
