Hermitian Quaternion Toeplitz Matrices by Quaternion-valued Generating Functions

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.