Trace Diagrams, Matrix Minors, and Determinant Identities

Details Trace Diagrams, Matrix Minors, and Determinant Identities Trace Diagrams, Matrix Minors, and Determinant Identities

2009-03-07

arxiv.org/abs/0903.1373v2

Involve Vol. 3 (2010), No. 1, 33-66

Mathematics

Combinatorics

This updated version includes significantly expanded proofs and a new generalization of the Jacobi determinant theorem

Scientific paper

10.2140/involve.2010.3.33

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.

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