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

This paper provides an introduction to trace diagrams at a level suitable for advanced undergraduates. Trace diagrams are a non-traditional notation for linear algebra. Vectors are represented by edges in a diagram, and matrices by markings along the edges of the diagram. The notation is rigorous and permits proofs more elegant than those written using traditional notation. We begin with the definition of trace diagrams, and move directly into two special cases that help orient the reader to the diagrammatic point-of-view. We then provide an explicit description of how they are calculated. Finally, we provide the diagrammatic perspective on some questions often posed by students seeing vectors and linear algebra for the first time. We also look at some questions inspired by the diagrammatic notation. We include several examples and exercises throughout, which are particularly important for adjusting to nonstandard notation., Comment: Several figures created using PGF/TikZ

This note concerns a one-line diagrammatic proof of the Cayley-Hamilton Theorem. We discuss the proof's implications regarding the "core truth" of the theorem, and provide a generalization. We review the notation of trace diagrams and exhibit explicit diagrammatic descriptions of the coefficients of the characteristic polynomial, which occur as the n+1 "simplest" trace diagrams. We close with a discussion of diagrammatic polarization related to the theorem.

Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein, and Vogt., Comment: Updated historical treatment of the subject. Figures drawn with PGF/TikZ; Handbook of Teichmuller Theory II, A. Papadopoulos (ed), EMS Publishing House, Zurich, 2008