High-level axioms for graphical linear algebra

Abstract

• We present useful symmetrical axioms for graphical linear algebra. • We use only the diagrammatic language and its associated reasoning principles. • We develop an approach to matrices, proving its equivalence to the classical one. We focus on a modular, graphical language—graphical linear algebra—and use it as high-level language for calculational reasoning. We propose a minimal framework of axioms that highlight the dualities and symmetries of linear algebra, and showcase the resulting diagrammatic calculus. Our work develops a relational approach to linear algebra, closely connected to classical relational algebra. With the symmetrical high-level axioms we are able to provide a fully diagrammatic proof that a fragment of Graphical Linear Algebra is equivalent to matrix algebra.