Trace diagram

In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation.

Let V be a vector space of dimension n over a field F (with n≥2), and let Fun(V,V) denote the linear transformations on V. An n-trace diagram is a graph ${\displaystyle {\mathcal {D}}=(V_{1}\sqcup V_{2}\sqcup V_{n},E)}$, where the sets V_i (i = 1, 2, n) are composed of vertices of degree i, together with the following additional structures:

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