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 similar 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 system arising from linear prediction of quaternion signals are employed to validate the efficiency of our proposed block diagonalized results. A numerical example of color video as third-order quaternion tensor is employed to validate the effectiveness of quaternion tensor singular value decomposition.

翻译：众所周知，复循环矩阵可由含有虚单位 $\mathtt{i}$ 的离散傅里叶矩阵对角化。本文主要证明：含有三个虚单位 $\mathtt{i}$、$\mathtt{j}$ 和 $\mathtt{k}$ 的离散四元数傅里叶矩阵无法对角化四元数循环矩阵。然而，通过置换离散四元数傅里叶变换矩阵，可将四元数循环矩阵分块对角化为1×1分块与2×2分块矩阵。借助该分块对角化形式，可类似复循环矩阵求逆的方式高效计算四元数循环矩阵的逆矩阵。我们利用该分块对角化形式研究元素为四元数的四元数张量的四元数张量奇异值分解。为验证所提分块对角化结果的有效性，本文采用四元数循环矩阵求逆及源于四元数信号线性预测的四元数Toeplitz系统求解等应用进行实验。此外，通过彩色视频的三阶四元数张量数值算例，验证了四元数张量奇异值分解的有效性。