Hermitian Quaternion Toeplitz Matrices by Quaternion-valued Generating Functions
Xue-lei Lin, Michael K. Ng, Junjun Pan
TL;DR
The paper extends Toeplitz spectral theory to Hermitian quaternion Toeplitz matrices by introducing a quaternion-valued generating function and a 2-by-2 block representation, establishing a Grenander–Szegö-type spectral distribution via the functions $\check{f}$ and $\hat{f}$ and the matrix $G[f](x)$. It then develops Strang's circulant preconditioners for quaternion Toeplitz systems, showing diagonalization through the quaternion discrete Fourier transform and providing precise eigenvalue mappings through the partial sums $f_m$. The authors prove clustering of the preconditioned spectrum near 1 and derive a superlinear convergence rate for the circulant-preconditioned conjugate gradient method, with $O(n\log n)$ per-iteration cost thanks to FFT-based Diagonalization. Numerical experiments in quaternion signal processing confirm faster convergence and improved accuracy of the preconditioned method compared to unpreconditioned CG, corroborating the theory. Overall, the work delivers a rigorous quaternionic extension of Grenander–Szegö theory and practical preconditioning techniques for efficient quaternion Toeplitz solvers.
Abstract
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-Szegö 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.