Fifty Three Matrix Factorizations: A Systematic Approach

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