Block-Diagonalization of Quaternion Circulant Matrices with Applications

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1429-1454, September 2024.
Abstract. It is well known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit [math]. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units [math], [math], and [math]. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similarly to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications, including computing the inverse of a quaternion circulant matrix and solving quaternion Toeplitz systems arising from linear prediction of quaternion signals, are employed to validate the efficiency of our proposed block- diagonalized results.