Efficient algorithms for the symbolic computation of matrices

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Symbolic computation (also called computer algebra) is a relatively recent research area in computer science and mathematics, which refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects, in particular, mathematical computations performed symbolically rather than numerically. Because of its lack of numerical errors, symbolic computation is widely used in mathematical research, engineering, and science. The overarching goal of this grant proposal is the design, analysis and implementation of efficient algorithms for matrices and tensors over a number of important algebraic structures in symbolic computation, such as computing various generalized inverses and normal forms, rank decomposition, simultaneous decomposition and solving matrix equations over various fields and polynomial rings. Their study is motivated by numerous applications in control theory, machine learning, data science, engineering, signal and color image processing, statistics, neural network, etc. For matrices over some fields, above questions have been explored in the past decades. Recently efficient symbolic algorithms for matrices over quaternion and generalized polynomials have attracted more and more attentions, in particular, these questions for tensors. We plan to employ some symbolic computation techniques such as Groebner bases to investigate these questions. We expect that this will lead to more efficient algorithms for computing generalized inverses and solving matrix (tensor) equations and systems of differential and difference equations. This should provide both theoretical and practical improvements over currently known algorithms. Finally, the algorithmic advances of this proposal will be implemented in computer algebra software such as Maple and SAGE. The projects all involve extensive training of highly qualified personnel (HQP) for their future positions in academia and industry.