We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from non-commutativity of quaternion multiplication, and due to that $\mu^2 = -1$ possesses infinite solutions in the quaternion domain. Handling of quaternionic matrices is consequently complicated in several aspects (definition of eigenstructure, determinant, etc.). Our research findings clarify the relation of the Quaternion Fourier Transform matrix to the standard (complex) Discrete Fourier Transform matrix, and the extend on which well-known complex-domain theorems extend to quaternions. We focus especially on the relation of Quaternion Fourier Transform matrices to Quaternion Circulant matrices (representing quaternionic convolution), and the eigenstructure of the latter. A proof-of-concept application that makes direct use of our theoretical results is presented, where we present a method to bound the Lipschitz constant of a Quaternionic Convolutional Neural Network. Code is publicly available at: \url{https://github.com/sfikas/quaternion-fourier-convolution-matrix}.

翻译：本文研究了傅里叶变换与卷积运算的四元数版本的矩阵形式。四元数作为一种强大的表示单元，其应用却面临诸多困难，主要源于四元数乘法的非交换性，以及由此导致的方程 $\mu^2 = -1$ 在四元数域内拥有无穷多解。因此，四元数矩阵的处理在多个方面（如特征结构、行列式等的定义）变得复杂。我们的研究成果阐明了四元数傅里叶变换矩阵与标准（复数）离散傅里叶变换矩阵之间的关系，并明确了哪些著名的复数域定理可以推广到四元数情形。我们特别关注四元数傅里叶变换矩阵与四元数循环矩阵（代表四元数卷积）之间的关系，以及后者的特征结构。作为对我们理论成果的直接应用，我们提出了一个概念验证应用，展示了一种估计四元数卷积神经网络 Lipschitz 常数上界的方法。代码已公开于：\url{https://github.com/sfikas/quaternion-fourier-convolution-matrix}。