Arithmetic, geometric, and harmonic means for accretive-dissipative matrices

TL;DR

This paper introduces and studies arithmetic, geometric, and harmonic means for accretive-dissipative matrices within an extended Loewner order, establishing inequalities and relations among these means.

Contribution

It defines new matrix means for accretive-dissipative matrices and proves fundamental inequalities and relations, extending classical matrix mean concepts.

Findings

Established AM-GM-HM inequality for accretive-dissipative matrices.

Compared harmonic mean and parallel sum, revealing their relation.

Provided examples illustrating the properties and inequalities.

Abstract

The concept of Loewner (partial) order for general complex matrices is introduced. After giving the definition of arithmetic, geometric, and harmonic mean for accretive-dissipative matrices, we study their basic properties. An AM-GM-HM inequality is established for two accretive-dissipative matrices in the sense of this extended Loewner order. We also compare the harmonic mean and parallel sum of two accretive-dissipative matrices, revealing an interesting relation between them. A number of examples are included.