Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai (2016) Matrix positivity preservers in fixed dimension. Comptes Rendus Mathématique, 354 (2). pp. 143-148. ISSN 1631-073X
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Abstract
A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomial case. As consequences, we derive tight linear matrix inequalities for Hadamard powers of positive semidefinite matrices, and a sharp asymptotic bound for the matrix cube problem involving Hadamard powers.