An Oppenheim type inequality for positive definite block matrices

TL;DR

This paper extends an Oppenheim type inequality for positive definite block matrices, removing the previous requirement that two matrices commute, thus broadening its applicability.

Contribution

It provides a new, more general Oppenheim type inequality for positive definite block matrices without the commutativity condition.

Findings

Established a determinantal inequality for positive definite block matrices

Removed the commutativity requirement present in previous results

Extended the applicability of Oppenheim type inequalities

Abstract

We present an Oppenheim type determinantal inequality for positive definite block matrices. Recently, Lin [Linear Algebra Appl. 452 (2014) 1--6] proved a remarkable extension of Oppenheim type inequality for block matrices, which solved a conjecture of G\"{u}nther and Klotz. There is a requirement that two matrices commute in Lin's result. The motivation of this paper is to obtain another natural and general extension of Oppenheim type inequality for block matrices to get rid of the requirement that two matrices commute.