四元数循环矩阵的块对角化及其应用
Block Diagonalization of Quaternion Circulant Matrices with Applications
AI总结：
本文证明四元数循环矩阵无法被离散四元数傅里叶矩阵对角化，但可通过置换的离散四元数傅里叶变换矩阵块对角化，并应用于高效求逆、求解Toeplitz系统及四元数张量奇异值分解。
摘要：
众所周知，复数循环矩阵可以通过带有虚数单位 $\mathtt{i}$ 的离散傅里叶矩阵进行对角化。本文的主要目的是证明四元数循环矩阵不能通过带有三个虚数单位 $\mathtt{i}$、$\mathtt{j}$ 和 $\mathtt{k}$ 的离散四元数傅里叶矩阵进行对角化。相反，四元数循环矩阵可以通过置换的离散四元数傅里叶变换矩阵块对角化为1×1块和2×2块矩阵。利用这种块对角化形式，可以类似于复数循环矩阵的求逆方式，高效地确定四元数循环矩阵的逆。我们利用这种块对角化形式来研究条目为四元数的四元数张量的四元数张量奇异值分解。包括计算四元数循环矩阵的逆以及求解由四元数信号线性预测产生的四元数Toeplitz系统在内的应用被用来验证我们提出的块对角化结果的高效性。一个彩色视频作为三阶四元数张量的数值示例被用来验证四元数张量奇异值分解的有效性。
英文摘要：
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.
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