In linear algebra applications, elementary matrices hold a significant role. This paper presents a diagrammatic representation of all $2^m\times 2^n$-sized elementary matrices in algebraic ZX-calculus, showcasing their properties on inverses and transpose through diagrammatic rewriting. Additionally, the paper uses this representation to depict the Jozsa-style matchgate in algebraic ZX-calculus. To further enhance practical use, we have implemented this representation in \texttt{discopy}. Overall, this work sets the groundwork for more applications of ZX-calculus such as synthesising controlled matrices [arXiv:2212.04462] in quantum computing.

翻译：在线性代数应用中，初等矩阵扮演着重要角色。本文提出了一种在代数ZX演算中表示所有$2^m\times 2^n$规模初等矩阵的图解方法，并通过图解重写展示了其逆矩阵与转置矩阵的性质。此外，本文利用该表示方法在代数ZX演算中描述了Jozsa型匹配门。为增强实际应用，我们已在\texttt{discopy}中实现了该表示方法。总体而言，本工作为ZX演算的更多应用（如量子计算中的受控矩阵合成[arXiv:2212.04462]）奠定了基础。