  We introduce meta-factorization, a theory that describes matrix
decompositions as solutions of linear matrix equations: the projector and the
reconstruction equation. Meta-factorization reconstructs known factorizations,
reveals their internal structures, and allows for introducing modifications, as
illustrated with SVD, QR, and UTV factorizations. The prospect of
meta-factorization also provides insights into computational aspects of
generalized matrix inverses and randomized linear algebra algorithms. The
relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m
method, and the CUR decomposition are revealed here as an illustration.
Finally, meta-factorization offers hints on the structure of new factorizations
and provides the potential of creating them.
