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The original geometric mean of two positive definite operators A and B is given by A#B = A1/2(A−1/2BA−1/2)1/2A1/2. In this article we provide a new proof to construct from the two-variable geometric mean to the multivariable mean via symmetrization introduced by Lawson and Lim [5]. Finally we provide an algorithm to find threevariable geometric mean via symmetrization, which plays an important role to construct higher-order geometric means. AMS Mathematics Subject Classification : 47B65; 47H07.

A NEW PROOF TO CONSTRUCT MULTIVARIABLE GEOMETRIC MEANS BY SYMMETRIZATION<xref>†</xref>