A complete solution of the permutability problem for Toeplitz and Hankel matrices

Conditions under which two Toeplitz matrices commute are known since at least 1998. For a pair of Hankel matrices, the corresponding commutativity conditions were recently obtained by Gel'fgat. In this paper, we complete our analysis of the much more complex problem of characterizing matrix pairs (T,H) such that T is a Toeplitz matrix, H is a Hankel matrix, and TH=HT. The only case left open in our previous publications on this subject is the one where the Toeplitz matrix T is centrosymmetric. Here, we present an exhaustive study of this case, which yields a complete solution of the above commutativity problem.