Negativity-preserving transforms of tuples of symmetric matrices

2023/10/27 by Belton, Alexander, Guillot, Dominique, Khare, Apoorva +1 · 3 citations
#15A18 #15B48 (primary) #26A48 #32A05 (secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2310.18041

Abstract

Compared to the entrywise transforms which preserve positive semidefiniteness, those leaving invariant the inertia of symmetric matrices reveal a surprising rigidity. We first obtain the classification of negativity preservers by combining recent advances in matrix analysis with some novel arguments using well chosen test matrices. We continue with the analogous classification in the multi-variable setting, revealing a striking separation of variables, with absolute monotonicity on one side and homotheties on the other. We conclude with the complex analogue of this result.

Cited by

• The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings
• Cholesky decomposition for symmetric matrices, Riemannian geometry, and random matrices
• Entrywise transforms preserving matrix positivity and non-positivity

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