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Since positive definite Hermitian matrices have become fundamental objects in many areas, a variety of theoretical and computational research topics have been arisen. Especially, the average of positive definite matrices is a very important notion to see the central tendency of objects. There are many different kinds of averages for a finite number of positive definite matrices such as quasi-arithmetic means, power means and Cartan barycenters. We generalize these averages to the setting of positive definite matrices equipped with probability measures of compact support, and show the monotonicity of quasi-arithmetic means for parameters≥  1{\geq 1}   , and connections with inequalities between quasi-arithmetic means and power means, and between quasi-arithmetic means and Cartan barycenters.

The quasi-arithmetic means and Cartan barycenters of compactly supported measures