Finding positive matrices subject to linear restrictions

We characterize the existence of a positive definite l×l matrix X the entries of which satisfy n nonhomogeneous linear conditions by the existence of a minimum for an associated function V, smooth and strictly convex on Rn. If there exist solutions X>0, then lim‖x‖→∞V(x)=+∞ and the critical point x0 of V can be approximated by the conjugate gradients method. Knowing x0 provides, by a simple analytic formula, the unique solution X maximizing the entropy (where λ1,…,λl are the eigenvalues of X) subject to the given restrictions. Related results are obtained in the semipositive definite case, too.