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.

Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs.

Let V be a vector space of dimension n over a field F (with n≥2), and let Hom(V,V) denote the linear transformations on V. An n-trace diagram is a graph

The "graph" underlying a trace diagram may have the following special features, which are not always included in the standard definition of a graph: Every framed trace diagram corresponds to a multilinear function between tensor powers of the vector space V. The degree-1 vertices correspond to the inputs and outputs of the function, while the degree-n vertices correspond to the generalized Levi-Civita symbol (which is an anti-symmetric tensor related to the determinant).

If a diagram has no output strands, its function maps tensor products to a scalar.

If there are no degree-1 vertices, the diagram is said to be closed and its corresponding function may be identified with a scalar.

By definition, a trace diagram's function is computed using signed graph coloring.

The scalar triple product identity follows because each is a different representation of the same diagram's function.

is used to indicate equality up to a scalar factor that depends only on the dimension n of the underlying vector space.

If a closed trace diagram is labeled by k different matrices, it may be interpreted as a function from

Trace diagrams may be specialized for particular Lie groups by altering the definition slightly.

In this context, they are sometimes called birdtracks, tensor diagrams, or Penrose graphical notation.

Trace diagrams have primarily been used by physicists as a tool for studying Lie groups.

The most common applications use representation theory to construct spin networks from trace diagrams.