The number of invariant polynomials of a matrix with prescribed off-diagonal blocks

Let A be an n x n matrix over an arbitrary field of the form A = [A(1,1)/A(2,1) A(1,2)/A(2,2)] where A(1,1) is an element of F-pxp, A(2,2) is an element of F-qxq , and p + q = n. We characterize the possible number of nontrivial invariant polynomials of A, when the submatrices A(1,2) and A(2,1) are prescribed. (C) 1997 Elsevier Science Inc., 1997.