Jordan form of the BLOCK Frobenius matrix and its inverse

1) -It is well known that Jordan form of the SCALAR Frobenius matrix is
Vendermonde matrix (single eigenvalues) or confluent V. matrix for
non-single eig.
-Inverse of the calssic Vandermonde consist from basic symmetrical
polynomials, for confluent Vand. is a little bit more sophysticated, but
known.

2) Now let's take into the accound the BLOCK Frobenius matrix. (Frobenius with sub-matrices)
Anyone know whether for such a matrix the Jordan form and its inverse is known in the literature ? Is it possible to calculate it in a general analytical form ? Especially the inverse ?

John .