Efficient algorithms for the symbolic computation of matrices

符号计算（也称为计算机代数）是计算机科学和数学中一个相对较新的研究领域，它指的是研究和开发用于操纵数学表达式和其他数学对象的算法和软件，特别是以符号而不是数字方式进行的数学计算。由于符号计算没有数值误差，因此在数学研究、工程和科学中得到了广泛的应用。本课题的总体目标是设计、分析和实现符号计算中一些重要代数结构上矩阵和张量的有效算法，如计算各种广义逆和范式、秩分解、联立分解和求解各种域和多项式环上的矩阵方程。他们的研究受到控制理论、机器学习、数据科学、工程、信号和彩色图像处理、统计学、神经网络等领域的众多应用的推动。对于某些领域的矩阵，上述问题在过去的几十年里已经得到了探讨。近年来，四元数和广义多项式矩阵的高效符号算法，特别是张量矩阵的高效符号算法受到越来越多的关注。我们计划采用一些符号计算技术，如Groebner基来研究这些问题。我们期望这将导致更有效的算法来计算广义逆和求解矩阵（张量）方程以及微分和差分方程系统。这应该为目前已知的算法提供理论和实践上的改进。最后，本文的算法进步将在Maple和SAGE等计算机代数软件中实现。这些项目都涉及对高素质人才（HQP）的广泛培训，为他们未来在学术界和工业界的职位做好准备。

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.