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.

翻译：本文研究了由四元数值函数生成的埃尔米特四元数托普利茨矩阵。我们证明此类生成函数必须是一个实值函数与一个具有虚分量的奇函数之和。这一设定不同于仅由实值函数生成的埃尔米特复数托普利茨矩阵情形。通过利用四元数矩阵的2×2分块复数表示，我们给出了四元数版本的格伦南德-塞格定理，该定理通过生成函数描述了埃尔米特四元数托普利茨矩阵的特征值分布。作为应用，我们研究了四元数信号处理中出现的埃尔米特四元数托普利茨线性系统的Strang循环预处理子。我们证明Strang循环预处理子可由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无法通过此类矩阵对角化。同时，我们验证了Strang循环预处理共轭梯度法在求解埃尔米特四元数托普利茨系统时的理论与数值收敛结果。