MATRICES WITH PRESCRIBED EIGENVALUES AND BLOCKS

Let F be a field and let n, p1, p2, p3 be positive integers such that n = p1 + p2 + p3. Let A = [ A(1,1) A(1,2) A(1,3) A(2,1) A(2,2) A(2,3) A(3,1) A(3,2) A(3,3) ] epsilon F-nxn, where the blocks A(i,) (j) epsilon F-pi (x pj) , i, j epsilon {1, 2, 3} and the blocks in the positions (i, j) are square. We study the possible eigenvalues of A, when A(1,2), A(1,3) and A(2,1) are fixed and the remaining blocks vary.