对偶复矩阵的低秩逼近
Low Rank Approximation of Dual Complex Matrices
摘要：
对偶复数可以表示2D空间中的刚体运动。对偶复矩阵与旋量理论联系在一起，在许多领域有着潜在的应用。本文研究了对偶复矩阵的低秩逼近问题。我们定义了对偶复向量的$2$ -范数和对偶复矩阵的Frobenius范数。这些范数是非负的对偶数。建立了对偶复矩阵的酉不变性。研究了平方对偶复矩阵的特征值，证明了一个$n \times n$对偶复埃尔米特矩阵恰有$n$特征值，且特征值是对偶数.给出了对偶复矩阵的奇异值分解定理，定义了对偶复矩阵的秩和可观秩，并研究了它们的性质。建立了对偶复矩阵的Eckart-Young类定理，并给出了一个用截断奇异值分解求对偶复矩阵低秩逼近的算法框架。对偶复矩阵的奇异值分解也为利用这些矩阵进行主成分分析提供了一个基本工具。数值实验报告。
英文摘要：
Dual complex numbers can represent rigid body motion in 2D spaces. Dual complex matrices are linked with screw theory, and have potential applications in various areas. In this paper, we study low rank approximation of dual complex matrices. We define $2$-norm for dual complex vectors, and Frobenius norm for dual complex matrices. These norms are nonnegative dual numbers. We establish the unitary invariance property of dual complex matrices. We study eigenvalues of square dual complex matrices, and show that an $n \times n$ dual complex Hermitian matrix has exactly $n$ eigenvalues, which are dual numbers. We present a singular value decomposition (SVD) theorem for dual complex matrices, define ranks and appreciable ranks for dual complex matrices, and study their properties. We establish an Eckart-Young like theorem for dual complex matrices, and present an algorithm framework for low rank approximation of dual complex matrices via truncated SVD. The SVD of dual complex matrices also provides a basic tool for Principal Component Analysis (PCA) via these matrices. Numerical experiments are reported.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
