Search for: All records

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 ofwhich we believe to be new. Our systematic approach, inspired by the generalized Cartan decom-position of the Lie theory, also encompasses known factorizations such as the SVD, the symmetriceigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factor-ization, and others thereby covering familiar matrix factorizations, as well as ones that were waitingto be discovered. We suggest that the Lie theory has one way or another been lurking hidden in thefoundations of the very successful field of matrix computations with applications routinely used in somany areas of computation. In this paper, we investigate consequences of the Cartan decompositionand the little known generalized Cartan decomposition for matrix factorizations. We believe thatthese factorizations once properly identified can lead to further work on algorithmic computationsand applications. 

The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules. 

We complete Dyson’s dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan’s symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra, reveal themselves to be perfectly valid coordinate systems that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decompositions. This, in turn, allows us to produce yet more random matrix theories from a single symmetric space. Yet, again, these random matrix theories arise from matrix factorizations, though ones that we are not aware have appeared in the literature.