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

Minghua Lin

Published 2012-06-02, updated 2016-01-21Version 3

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.