This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how to calculate them. The projectors can be applied to solve linear matrix equations, generate low-rank approximations, or design randomized matrix algorithms. But also, as demonstrated, they can be applied in cryptography to encrypt and decrypt messages. The paper discusses some of the security implications of this application and leaves some questions open for further investigation. The basic concepts are illustrated with source code listings. Finally, this work shares some personal reflections on the meaning and importance of understanding in the time of the artificial intelligence revolution.

翻译：本文探讨了矩阵分解与线性矩阵方程之间的关系。研究表明，每种矩阵分解都定义了两种隐藏的投影算子——一种针对矩阵的列空间，另一种针对行空间，并阐明了如何计算这些投影算子。这些投影算子可用于求解线性矩阵方程、生成低秩近似或设计随机矩阵算法。此外，本文还展示了它们如何在密码学中用于加密和解密信息。论文讨论了该应用的一些安全影响，并留下一些开放性问题供进一步研究。基本概念通过源代码列表加以说明。最后，本文分享了在人工智能革命时代关于理解的含义与重要性的一些个人思考。