  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.
