  This paper concerns the analysis of an unbounded Toeplitz-like operator
generated by a rational matrix function having poles on the unit circle T. It
extends the analysis of such operators generated by scalar rational functions
with poles on T found in [11,12,13]. A Wiener-Hopf type factorization of
rational matrix functions with poles and zeroes on T is proved and then used to
analyze the Fredholm properties of such Toeplitz-like operators. A formula for
the index, based on the factorization, is given. Furthermore, it is shown that
the determinant of the matrix function having no zeroes on T is not sufficient
for the Toeplitz-like operator to be Fredholm, in contrast to the classical
case.
