  There is interesting internal structure in the nullspaces of graph and
hypergraph adjacency matrices, especially for trees, bipartite graphs, and
related combinatorial classes. The zero loci of nullvectors, i.e., their zero
coordinates' indices, encode information about matchings, coverings, and edges'
influence on rank. This set system is the lattice of flats of a ``kernel
matroid'', a subsystem of which are the ``stalled'' sets closed under skew zero
forcing (SZF), a graph percolation/infection model known to have connections
with rank and nullity. For a wide variety of graphs, the lattice of SZF-closed
sets is also a matroid, a fact which can be used to obtain a polynomial-time
algorithm for computing the skew zero forcing number. This contrasts with the
general case, where we show that the corresponding decision problem is NP-hard.
We also define skew zero forcing for hypergraphs, and show that, for linear
hypertrees, the poset of SZF-closed sets is dual to the lattice of ideals of
the hypergraph's nullvariety; while, for complete hypergraphs, the SZF-closed
sets and the zero loci of nullvectors are more loosely related.
