MATRIX COMPLETIONS, NORMS, AND HADAMARD-PRODUCTS

Let M(m, n) (respectively, H(n)) denote the space of m x n complex matrices (respectively, n x n Hermitian matrices). Let S subset-of H(n) be a closed convex set. We obtain necessary and sufficient conditions for X0 is-an-element-of S to attain the maximum in the following concave maximization problem: max {lambda(min) (A + X): X is-an-element-of S} where A is-an-element-of H(n) is a fixed matrix. Let . denote the Hadamard (entrywise) product, i.e., given matrices A = [a(ij)] B = [b(ij)] is-an-element-of M(m, n) we define A . B [a(ij)b(ij)] is-an-element-of M(m, n). Using the necessary and sufficient conditions mentioned above we give elementary and unified proofs of the following results. (a) For any A is-an-element-of M(m, n) omega(A) = max{\x* Ax\:x is-an-element-of C(n), X*x = 1} less-than-or-equal-to 1 if and only if there is a matrix Z is-an-element-of H(n) such that (A*/I+Z I-Z/A) greater-than-or-equal-to 0. (b) For any A is-an-element-of M(m, n) max{\\A . B\\infinity: \\B\\infinity less-than-or-equal-to 1} less-than-or-equal-to 1 if and only if there are matrices P is-an-element-of H(m), Q is-an-element-of H(n) such that (A*/P Q/A) greater-than-or-equal-to 0, P . I less-than-or-equal-to I, Q . I less-than-or-equal-to I. (c) For any A is-an-element-of M(n, n) max{omega(A . B): omega(B) less-than-or-equal-to 1} less-than-or-equal-to 1 if and only if there is a matrix P is-an-element-of H(n) such that (A*/P P/A) greater-than-or-equal-to 0, P . I less-than-or-equal-to I. We also consider other norms that can be represented in this way and show that if a norm can be represented in this way then so can its dual.