Block-Diagonalization of Quaternion Circulant Matrices with Applications
Junjun Pan, Michael K. Ng
Abstract
.It is well known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit \(\mathtt{i}\). 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 \(\mathtt{i}\), \(\mathtt{j}\), and \(\mathtt{k}\). 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.Keywordscirculant matrixquaternionblock-diagonalizationdiscrete Fourier transformtensorsingular value decompositionMSC codes65F1097N3094A08