An Oppenheim type inequality for positive definite block matrices

Abstract

We present an Oppenheim type determinantal inequality for positive definite block matrices. Recently, Lin [Linear Algebra Appl. 2014;452:1–6] proved a remarkable extension of Oppenheim type inequality for block matrices, which solved a conjecture of Günther 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.

Keywords:

• Hadamard product
• Oppenheim's inequality
• Fischer's inequality
• positive definite
• block matrices

2010 Mathematics Subject Classifications:

• 15A45
• 15A60
• 47B65

Acknowledgments

We are thankful to anonymous reviewers for the helpful comments and suggestions. The author is deeply indebted to Minghua Lin for many useful discussions over the years as well as helpful comments on the manuscript. This work was supported by NSFC (Grant Nos. 11931002 and 11671124).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by NSFC [grant number 11931002], [grant number 11671124].