Negativity-preserving transforms of tuples of symmetric matrices

Compared to the entrywise transforms which preserve positive semidefiniteness, those leaving invariant the inertia of symmetric matrices reveal a surprising rigidity. We achieve the classification of negativity preservers by combining recent advances in matrix analysis with some elementary arguments using well chosen test matrices. We unveil a complete analysis of the multi-variable setting with a striking combination of absolute monotonicity and single-variable rigidity appearing in our conclusions.