  Trace diagrams are structured graphs with edges labeled by matrices. Each
diagram has an interpretation as a particular multilinear function. We provide
a rigorous combinatorial definition of these diagrams using a notion of signed
graph coloring, and prove that they may be efficiently represented in terms of
matrix minors. Using this viewpoint, we provide new proofs of several standard
determinant formulas and a new generalization of the Jacobi determinant
theorem.
