In this paper, we study the change of spectrum and the existence of Riesz bases of specific classes of n × n unbounded operator matrices, called: diagonally and off-diagonally generalized subordinate block operator matrices. An application to a n × n Gribov operator matrix acting on a sum of Bargmann spaces, illustrates the abstract results. As example, we consider a particular Gribov operator matrix by taking special values of the real parameters of Pomeron.