The condition number c sub phi of a nonsingular matrix A is defined by c sub phi (A) = phi (A) phi (A superscript -1) where ordinarily phi is a norm. It was shown by J.D. Riley that if A is positive definite, c sub phi (A + kI) = or <c sub phi (A) when k> 0 and phi squared (A) is the maximum eigenvalue of AA* or phi squared (A) = Tr AA*. In this paper it is shown more generally that c sub phi (A + B) = or <c sub phi (B) when phi satisfies phi (U) = or <phi (V) if V-U is positive definite and when A, B are positive definite satisfying c sub phi (A) = or <c sub phi (B). Some related inequalities are also obtained. As suggested by Riley, these results may be of practical use in solving a system Ax = d of linear equations when A is positive definite but ill-conditioned. (Author).