Representing and Implementing Matrices Using Algebraic ZX-calculus

Elementary matrices play an important role in linear algebra applications. In this paper, we develop and implement (in \texttt{discopy}) an algorithm to represent all the elementary matrices of size $2^m\times 2^m$ using algebraic ZX-calculus. Then we show their properties on inverses and transpose using rewriting rules of ZX-calculus. As a consequence, we are able to depict any matrices of size $2^m\times 2^n$ by string diagrams without resort to a diagrammatic normal form for matrices as shown in [arXiv:2007.13739]. We show how this representation method could be used for representing symmetrising projectors which are essential in AKLT states. By doing so we pave the way towards visualising by string diagrams important matrix technologies deployed in AI especially machine learning.