On a type of circulants

AbstractAccording to Aitken, a “retrocirculant” is the product of a circulant and a certain permutation matrix P. In [1], he determined the eigenvalues of the retrocirculants. In his proof, he used PΩ = ΩP where Ω is a certain unitary matrix. Here we present a necessary and sufficient condition for a permutation matrix to commute with Ω. We also determine the eigenvalues of many retrocirculants