Matrix factorization in dual number algebra, a hypercomplex system, has been applied to kinematics, mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-$k$ approximation under a newly defined Froubenius norm in dual complex number system. The CDSVD is also related to the dual Moore-Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate the less computational cost of our proposed CDSVD. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain fMRI data and then identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function.

翻译：对偶代数是超复数系统的一种，其矩阵分解方法已被应用于运动学、机构学等领域。本文基于对偶矩阵的奇异值分解，提出了一种识别大脑行波等时空模式的方法。理论上，我们提出了具有显式表达式的对偶复矩阵的紧致对偶奇异值分解（CDSVD），并给出了其存在的充要条件。进一步地，基于CDSVD，我们在新定义的对偶复数系弗罗贝尼乌斯范数下，给出了最佳秩-k近似的优解。CDSVD还与对偶摩尔-彭罗斯广义逆相关。数值实验中，我们与其他现有算法进行了对比，结果表明我们提出的CDSVD计算成本更低。接着，我们使用模拟时间序列数据和道路监控视频进行实验，论证了对偶矩阵无穷小部分在时空模式识别中的有益效果。最后，我们将该方法应用于大规模脑功能磁共振成像数据，识别出三种行波，并进一步验证了分析结果与当前大脑皮层功能认知的一致性。