Matrix decompositions in dual number representations have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices, specifically geared towards its application in traveling wave identification, utilizing the concept of proper orthogonal decomposition. When dealing with large-scale problems, we present explicit solutions for the QR, thin QR, and randomized QR decompositions of dual number matrices, along with their respective algorithms with column pivoting. The QR decomposition of dual matrices is an accurate first-order perturbation, with the Q-factor satisfying rigorous perturbation bounds, leading to enhanced orthogonality. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Subsequently, we employed the QR decomposition of dual matrices to compute the DMPGI, thereby attaining results of higher precision. Moreover, we apply the QR decomposition in the context of traveling wave identification, employing the notion of proper orthogonal decomposition to perform a validation analysis of large-scale functional magnetic resonance imaging (fMRI) data for brain functional circuits. Our approach significantly improves the identification of two types of wave signals compared to previous research, providing empirical evidence for cognitive neuroscience theories.

翻译：对偶数表示下的矩阵分解近年来在运动学和计算机图形学等领域发挥了重要作用。本文提出了一种针对对偶数矩阵的QR分解算法，并利用本征正交分解的概念，专门应用于行波识别。在处理大规模问题时，我们给出了对偶数矩阵的QR、瘦QR以及随机化QR分解的显式解，并提供了相应的带列选主元算法。对偶矩阵的QR分解是一种精确的一阶扰动，其中Q因子满足严格的扰动界，从而增强了正交性。在数值实验中，我们讨论了不同QR算法在面对各种大规模对偶矩阵时的适用性，并给出了各自的应用范围。随后，我们利用对偶矩阵的QR分解计算DMPGI，从而获得了更高精度的结果。此外，我们将QR分解应用于行波识别的背景中，借助本征正交分解的概念，对大脑功能回路的大规模功能磁共振成像（fMRI）数据进行了验证分析。与以往研究相比，我们的方法显著提高了两类波信号的识别能力，为认知神经科学理论提供了经验证据。