Matrix positivity preservers in fixed dimension

Alexander Belton (Lancaster University)

Tuesday 19th April, 2016 16:00-17:00 Maths 522


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 is a simple consequence that every analytic function with non-negative Maclaurin coefficients, when applied entrywise, preserves positive semidefiniteness of matrices of any order. The converse is also true: as Schoenberg proved, any 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 how large this term may be.

(Joint work with Dominique Guillot, Apoorva Khare and Mihai Putinar)

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