The matrix weighted average (H = V^-1)^-1Hb, where H and V are symmetric positive definite matrices and b is a vector, is shown to lie in one ellipsoid if V is bounded from below, V "* @< V, another ellipsoid if V is bounded from above, V @< V*, and another ellipsoid if V is bounded from above and below, V "*@< V @< V*. These results are applied to bound the posterior mean vector of the normal linear regression model.