A Not-so-Characteristic Equation: the Art of Linear Algebra

TL;DR

This paper introduces trace diagrams, a diagrammatic technique that simplifies and unifies core concepts in linear algebra such as determinants, traces, and characteristic polynomials, revealing their interconnectedness.

Contribution

It presents a novel diagrammatic approach to linear algebra that naturally derives key results and clarifies their underlying structure.

Findings

Trace diagrams provide elegant solutions to linear algebra questions.

Standard linear algebra constructs arise naturally from the diagrams.

The approach reveals the fundamental connections between different linear algebra concepts.

Abstract

Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from linear algebra that offers elegant solutions to all these questions. The doodles, known as trace diagrams, are graphs labeled by matrices which have a correspondence to multilinear functions. This correspondence permits computations in linear algebra to be performed using diagrams. The result is an elegant theory from which standard constructions of linear algebra such as the determinant, the trace, the adjugate matrix, Cramer's rule, and the characteristic polynomial arise naturally. Using the diagrams, it is easy to see how little structure gives rise to these various results, as they all can be `traced' back to the definition of the determinant and inner product.