  We establish a connection between problems studied in rigidity theory and
matroids arising from linear algebraic constructions like tensor products and
symmetric products. A special case of this correspondence identifies the
problem of giving a description of the correctable erasure patterns in a
maximally recoverable tensor code with the problem of describing bipartite
rigid graphs or low-rank completable matrix patterns. Additionally, we relate
dependencies among symmetric products of generic vectors to graph rigidity and
symmetric matrix completion. With an eye toward applications to computer
science, we study the dependency of these matroids on the characteristic by
giving new combinatorial descriptions in several cases, including the first
description of the correctable patterns in an (m, n, a=2, b=2) maximally
recoverable tensor code.
