It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.