Theoretical Statistics and Mathematics Unit, ISI Delhi

The matrix geometric mean

by Rajendra Bhatia and Rajeeva L. Karandikar

An attractive candidate for the geometric mean of $m$
positive definite matrices $A_1,\ldots , A_m$ is their Riemannian
barycentre $G$. One of its important properties, monotonicity in the
$m$ arguments, has been established recently by J. Lawson and Y. Lim.
We give a much simpler proof of this result, and prove some other
inequalities. One of these says that, for every unitarily invariant
norm, $||| G |||$ is not bigger than the geometric mean of $|||A_1|||,
\ldots, |||A_m|||$.

isid/ms/2011/02 [fulltext]

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