Uniqueness of the solutions of some completion problems Original Research Article

We determine the conditions for uniqueness of the solutions of several completion problems including the positive semi-definite completion, the distance matrix completion, and the contractive matrix completion. The conditions depend on the existence of a positive semi-definite matrix satisfying certain linear constraints. Numerically, such conditions can be checked by existing computer software such as semi-definite programming routines. Some numerical examples are given to illustrate the results, and show that some results in a recent paper of Alfakih are incorrect. Discrepancies in the proof of Alfakih are explained.