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Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2020

Schur polynomials and matrix positivity preservers

Abstract

A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to 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 preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.
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Dates and versions

hal-02173745 , version 1 (04-07-2019)

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Alexander Belton, Dominique Guillot, Apoorva Khare, Mihai Putinar. Schur polynomials and matrix positivity preservers. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6408⟩. ⟨hal-02173745⟩
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