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Schur polynomials and matrix positivity preservers

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.

Publication metadata

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