Quaternion Toeplitz matrices and their fundamental properties
Muhammad Ahsan Khan, Sohail Khan
TL;DR
This work extends Toeplitz matrix theory to quaternion-valued entries, addressing how noncommutativity affects structure and classification. It develops a shift-based framework for quaternion Toeplitz matrices, establishes a decomposition T = T(p,ψ) + p0I and a key T-ΓTΓ* characterization, and analyzes products and normality within the commutative-quaternion setting. The main contributions include a precise condition for the product of quaternion Toeplitz matrices to remain Toeplitz and a complete normality classification for Toeplitz matrices with commuting quaternion entries, providing foundational tools for quaternion Toeplitz algebra and potential applications in noncommutative operator theory. These results lay groundwork for further exploration of quaternionic structured matrices in analysis and applications where noncommutative scalars arise.
Abstract
Toeplitz matrices are characterized by their constant diagonals, have been extensively studied in various settings, including over real and complex numbers. However, their study over quaternions is quite sparse. In this paper, we investigate the structure and the algebraic properties of quaternion Toeplitz matrices. Most importantly, we established a complete characterization of all normal Toeplitz matrices having entries commutative quaternions.