标题： 厄米特四元数托普利茨矩阵由四元数值生成函数 显示英文标题

标题： Hermitian Quaternion Toeplitz Matrices by Quaternion-valued Generating Functions

摘要： 在本文中，我们研究由四元数值函数生成的厄米特四元数托普利茨矩阵。我们证明，这样的生成函数必须是实值函数和具有虚部的奇函数的和。这种设定与仅由实值函数生成的厄米特复托普利茨矩阵的情况不同。通过使用四元数矩阵的2x2块复表示，我们给出了一个四元数版本的Grenander-Szegö定理，该定理用其生成函数来描述厄米特四元数托普利茨矩阵的特征值分布。作为应用，我们研究了来自四元数信号处理的厄米特四元数托普利茨线性系统的Strang循环预条件子。我们证明，Strang的循环预条件子可以通过离散四元数傅里叶变换矩阵对角化，而一般的四元数循环矩阵则不能通过它们对角化。此外，我们验证了用于求解厄米特四元数托普利茨系统的Strang循环预条件共轭梯度法的理论和数值收敛结果。 显示英文摘要

摘要： In this paper, we study Hermitian quaternion Toeplitz matrices generated by quaternion-valued functions. We show that such generating function must be the sum of a real-valued function and an odd function with imaginary component. This setting is different from the case of Hermitian complex Toeplitz matrices generated by real-valued functions only. By using of 2-by-2 block complex representation of quaternion matrices, we give a quaternion version of Grenander-Szeg\"{o} theorem stating the distribution of eigenvalues of Hermitian quaternion Toeplitz matrices in terms of its generating function. As an application, we investigate Strang's circulant preconditioners for Hermitian quaternion Toeplitz linear systems arising from quaternion signal processing. We show that Strang's circulant preconditioners can be diagionalized by discrete quaternion Fourier transform matrices whereas general quaternion circulant matrices cannot be diagonalized by them. Also we verify the theoretical and numerical convergence results of Strang's circulant preconditioned conjugate gradient method for solving Hermitian quaternion Toeplitz systems.