Matrix Decompositions of Dual Matricesand Their Applications toTraveling Wave Identification in the Brain

江苏省应用数学（中国矿业大学）中心系列学术报告

报告题目：Matrix Decompositions of Dual Matricesand Their Applications toTraveling Wave Identification in the Brain

报告人：魏益民 教授 单位： 复旦大学数学学院

报告时间：2026年4月22日（周三）晚上20:00-21:00

腾讯会议：577-134-265

https://meeting.tencent.com/dm/ebkTh8nRErtS

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数学学院

报告人及报告内容简介：

魏益民于1997年在复旦大学数学研究所获得理学博士学位。他是复旦大学数学学院的教授和博士生导师、上海市应用数学重点实验室的研究人员，曾获得上海市高校优秀青年教师和上海市“曙光”学者称号，并获得上海市自然科学二等奖、三等奖。他多次入选爱思唯尔“中国高被引学者”榜单。‌他在国际权威期刊如SIAM Journal on Matrix Analysis and Applications、IEEE Transactions on Automatic Control等发表了150余篇论文，并出版了多部英文专著。他的Google学术引用次数超过12000次，H指数为55。

Abstract: Matrix factorizations in dual number algebra, a hypercomplex number system, havebeen applied to kinematics, spatial mechanisms, and other fields recently. We develop an approachto identify spatiotemporal patterns in the brain such as traveling waves using the singular valuedecomposition (SVD) of dual matrices in this paper. Theoretically, we propose the compact dualsingular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well asa necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we reporton the optimal solution to the best rank-k approximation under a newly defined quasi-metric in thedual complex number system. The CDSVD is also related to the dual Moore--Penrose generalizedinverse. Numerically, comparisons with other available algorithms are conducted, which indicate lesscomputational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD canidentify the true rank of the original matrix from the noise-added matrix, but the classical SVDcannot. Next, we employ experiments on simulated time-series data and a road monitoring video todemonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal patternidentification. Finally, we apply this approach to the large-scale brain functional magnetic resonanceimaging data, identify three kinds of traveling waves, and further validate the consistency betweenour analytical results and the current knowledge of cerebral cortex function.