Completely strong superadditivity of generalized matrix functions

TL;DR

This paper proves a strong superadditivity inequality for generalized matrix functions over positive semidefinite matrices, extending previous results and providing a new proof for Thompson's classical determinant inequality.

Contribution

It establishes a comprehensive superadditivity property for generalized matrix functions, broadening the understanding of matrix inequalities and extending prior work.

Findings

Generalized matrix functions satisfy block-matrix strong superadditivity.

The result extends previous inequalities by Paksoy, Turkmen, and Zhang.

A new, concise proof of Thompson's determinant inequality is provided.

Abstract

We prove that generalized matrix functions satisfy a block-matrix strong superadditivity inequality over the cone of positive semidefinite matrices. Our result extends a recent result of Paksoy-Turkmen-Zhang (V. Paksoy, R. Turkmen, F. Zhang, Inequalities of generalized matrix functions via tensor products, Electron. J. Linear Algebra 27 (2014) 332-341.). As an application, we obtain a short proof of a classical inequality of Thompson (1961) on block matrix determinants.