Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

Abstract

In this paper we prove among others that, if (Aj)j=1,...,m are positive definite matrices of order n ≥ 2 and qj ≥ 0, j = 1, ..., m with ∑j=1mqj=1$$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$, then 0≤11−mini∈{1,…,m}{qi}×[∑​i=1mqi(1−qi)[det(Ai)]−1−2n+1∑​1≤i<j≤mqiqj[det(Ai+Aj)]−1]≤∑​i=1mqi[det(Ai)]−1−[det(∑​i=1mqiAi)]−1≤1mini∈{1,…,m}{qi}×[∑​i=1mqi(1−qi)[det(Ai)]−1−2n+1∑​1≤i<j≤mqiqj[det(Ai+Aj)]−1].

Keywords

• positive definite matrices
• determinants
• inequalities
• 47a63
• 26d15
• 46c05