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

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.