The paper gives a complete classification of entrywise sign preservers of positive definiteness for fixed matrix dimensions over the reals and complexes.
Negativity-preserving transforms of tuples of symmetric matrices
2 Pith papers cite this work. Polarity classification is still indexing.
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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 relying on well chosen test matrices, Sidon sets from number theory, and analytic properties of absolutely monotone functions. We continue with the analogous classification in the multi-variable setting, revealing for the first time a striking separation of variables, with absolute monotonicity on one side and only homotheties on the other. We conclude with the complex analogue of this result.
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A survey of dimension-free entrywise positivity preservers with links to metric embeddings, Schur polynomials, finite fields, and an appendix on sphere packings via Schoenberg's theorem.
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Entrywise transforms preserving matrix positivity and non-positivity
The paper gives a complete classification of entrywise sign preservers of positive definiteness for fixed matrix dimensions over the reals and complexes.
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The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings
A survey of dimension-free entrywise positivity preservers with links to metric embeddings, Schur polynomials, finite fields, and an appendix on sphere packings via Schoenberg's theorem.