Matrix completions, norms and Hadamard products

Abstract

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 sufficient conditions for X 0 ∈ S to attain the maximum in the following concave maximization problem: max {λ min (A + X): X ∈ S} where A ∈ H n is a fixed matrix. Let o denote the Hadamard (entrywise) product, i.e., given matrices A = [a ij ], B = [b ij ] ∈ M m,n we define A o B = [a ij b ij ] ∈ M m,n . Using the necessary and sufficient conditions mentioned above we give elementaty and unified proofs of the following results