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Lookup NU author(s): Professor Mihai Putinar
This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).
Abstract. A classical result due to Schoenberg (1942) identifies all real-valued functions that preserve positivesemidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg’s work has continuedto attract significant interest, including renewed recent attention due to applications in high-dimensional statistics.However, despite a great deal of effort in the area, an effective characterization of entrywise functions preservingpositivity in a fixed dimension remains elusive to date. As a first step in this direction, we characterize new classesof polynomials preserving positivity in fixed dimension. As a consequence, we obtain novel tight linear matrix inequalitiesfor Hadamard powers of matrices. The proof of our main result is representation theoretic, and employsthe theory of Schur polynomials. An alternate, variational approach also leads to several interesting consequencesincluding (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connectionsbetween generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a novel description ofthe simultaneous kernels of Hadamard powers.
Author(s): Belton A, Guillot D, Khare A, Putinar M
Publication type: Article
Publication status: Published
Journal: Discrete Mathematics and Theoretical Computer Science
Year: 2020
Volume: DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
Online publication date: 22/04/2020
Acceptance date: 16/02/2016
Date deposited: 18/02/2016
ISSN (electronic): 1365-8050
Publisher: DMTCS
URL: https://dmtcs.episciences.org/6408
