The Hadamard product of two matrices is formed by multiplying corresponding entries, and the Schur product theorem states that this operation preserves positive semidefiniteness.

It follows immediately that every analytic function with non-negative Maclaurin coefficients, when applied entrywise, preserves positive semidefiniteness for matrices of any order. The converse is due to Schoenberg: a function which preserves positive semidefiniteness for matrices of arbitrary order is necessarily analytic and has non-negative Maclaurin coefficients.

For matrices of fixed order, the situation is more interesting. This talk will present recent work which shows the existence of polynomials with negative leading term which preserve positive semidefiniteness, and characterises precisely how large this term may be. (Joint work with D. Guillot, A. Khare and M. Putinar.)