  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.
