Matrix Completions , Norms and Hadamard Productsroy

Let M m;n (respectively, H n) denote the space of m n complex matrices (respectively, n n Hermitian matrices). Let S H n be a closed convex set. We obtain necessary and suucient conditions for X 0 2 S to attain the maximum in the following concave maximization problem: maxf min (A + X) : X 2 Sg where A 2 H n is a xed matrix. Let denote the Hadamard (entrywise) product, i.e., given matrices A = a ij ]; B = b ij ] 2 M m;n we deene A B = a ij b ij ] 2 M m;n. Using the necessary and suucient conditions mentioned above we give elementary and uniied proofs of the following results.