  The success of matrix factorizations such as the singular value decomposition
(SVD) has motivated the search for even more factorizations. We catalog 53
matrix factorizations, most of which we believe to be new.
  Our systematic approach, inspired by the generalized Cartan decomposition of
Lie theory, also encompasses known factorizations such as the SVD, the
symmetric eigendecomposition, the CS decomposition, the hyperbolic SVD,
structured SVDs, the Takagi factorization, and others thereby covering familiar
matrix factorizations as well as ones that were waiting to be discovered.
  We suggest that Lie theory has one way or another been lurking hidden in the
foundations of the very successful field of matrix computations with
applications routinely used in so many areas of computation. In this paper, we
investigate consequences of the Cartan decomposition and the little known
generalized Cartan decomposition for matrix factorizations.
  We believe that these factorizations once properly identified can lead to
further work on algorithmic computations and applications.
