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An Oppenheim type inequality for positive definite block matrices

Yongtao Li, Yuejian Peng

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 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.

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. 452 (2014) 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.

Paper Structure

This paper contains 3 sections, 11 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The base case $m=2$
  3. The General case

Key Result

Theorem 1.1

(see Lin14) Let ${\bm{A}} =[A_{ij}]_{i,j=1}^n$ and ${\bm{B}} =[B_{ij}]_{i,j=1}^n\in \mathbb{M}_n(\mathbb{M}_k)$ be positive definite matrices such that every $A_{ij}$ of ${\bm{A}}$ commutes with every $B_{rs}$ of ${\bm{B}}$. Then where ${\bm{A}}_{\mu}=[A_{ij}]_{i,j=1}^{\mu}$ and ${\bm{B}}_{\mu}=[B_{ij}]_{i,j=1}^{\mu}$ denote the $\mu \times \mu$ leading principal block submatrices of ${\bm{A}}$ a

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 1 more