  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.
