Hadamard duals, retractability and Oppenheim's inequality
Shaun Fallat, Charles R. Johnson · Operators and Matrices · 2007
Oppenheim's determinantal inequality was originally proved for positive semidefinite matrices and has produced many interesting consequences and applications. Positive semidefinite matrices were a natural class to consider partly because they are closed under Hadamard (or entry-wise) multiplication. Since Oppenheim's original contribution, others have considered similar inequalities for M -matrices, inverse M -matrices and totally nonnegative matrices. We attempt to unify many of these existing results dealing with Oppenheim's inequality, and our approach relies on two major themes: retractions and Hadamard duals. Retractions are a type of diagonal perturbation and the Hadamard dual is a maximal collection of matrices with a closure property under Hadamard multiplication. These notions are applied to yield results that generalize Oppenheim's original result.