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This paper explores the relationship between matrix factorizations and linearmatrix equations. It shows that every matrix factorization defines two hiddenprojectors, one for the column space and one for the row space of a matrix, andhow to calculate them. The projectors can be applied to solve linear matrixequations, generate low-rank approximations, or design randomized matrixalgorithms. But also, as demonstrated, they can be applied in cryptography toencrypt and decrypt messages. The paper discusses some of the securityimplications of this application and leaves some questions open for furtherinvestigation. The basic concepts are illustrated with source code listings.Finally, this work shares some personal reflections on the meaning andimportance of understanding in the time of the artificial intelligencerevolution.

The secret life of matrix factorizations: how matrix decompositions reveal and keep secrets of linear equations and what we can do about it