  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}.
