Revisiting a sharpened version of Hadamard's determinant inequality

TL;DR

This paper revisits and refines a version of Hadamard's determinant inequality for positive semidefinite matrices, providing a new proof, conditions for equality, and a block extension based on majorization and Thompson's result.

Contribution

It offers a new proof of a sharpened Hadamard inequality using majorization, and describes equality conditions and a block extension.

Findings

New proof of the inequality using majorization

Complete characterization of equality conditions

Extension to block matrices based on Thompson's result

Abstract

Hadamard's determinant inequality was refined and generalized by Zhang and Yang in [Acta Math. Appl. Sinica 20 (1997) 269-274]. Some special cases of the result were rediscovered recently by Rozanski, Witula and Hetmaniok in [Linear Algebra Appl. 532 (2017) 500-511]. We revisit the result in the case of positive semidefinite matrices, giving a new proof in terms of majorization and a complete description of the conditions for equality in the positive definite case. We also mention a block extension, which makes use of a result of Thompson in the 1960s.