[The matrix weighted average (H = V extasciicircum-1) extasciicircum-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 "* @ extless V, another ellipsoid if V is bounded from above, V @ extless V*, and another ellipsoid if V is bounded from above and below, V "*@ extless V @ extless V*. These results are applied to bound the posterior mean vector of the normal linear regression model.]