Minimizing the condition number of a positive definite matrix by completion

Summary.We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: $$\min\left\{ {\rm cond}\left(\mat{cc} A & B^{\rm H} \\ B & X \rix\right): \mat{cc} A & B^{\rm H} \\ B & X \rix \mbox {\rm positive definite} \right\},$$ \noindent where $A$ is an $n\times n$ Hermitian positive definite matrix, $B$ a $p\times n$ matrix and $X$ is a free $p\times p$ Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number.