Matrices That Commute With a Permutation Matrix
Let P be an n x n permutation matrix, and let p be the corresponding permutation. Let A be a matrix such that AP = PA. It is well known that when p is an n-cycle, A is permutation similar to a circulant matrix. We present results for the band patterns in A and for the eigenstructure of A when p consists of several disjoint cycles. These results depend on the greatest common divisors of pairs of cycle lengths.
Linear Algebra and Its Applications
Stuart, J. L.,
Weaver, J. R.
(1991). Matrices That Commute With a Permutation Matrix. Linear Algebra and Its Applications, 150, 255-265.
Available at: https://aquila.usm.edu/fac_pubs/6965