New geometries on positive-definite matrices and their relation to the power means – Bruno Iannazzo (Università degli Studi di Perugia)


Aula Magna – Dipartimento di Matematica.


We introduce a new family of geometries on the cone of positive definite matricesobtained from the Hessian of the power potential and provide explicit expressionsfor related quantities such as geodesics and distances.This generalizes in some sense the geometry obtained from the Hessian of thelogarithmic potential, whose geodesic has been understood as the weightedgeometric mean of two matrices. Indeed, the new geometries provide a definitionof matrix power mean alternative to the existing ones.We discuss some properties of the proposed power mean and relate it with thegeometric mean.Finally, we discuss on how these geometries can be used to accelerate theconvergence of optimization algorithms for the matrix geometric mean of severalvariables.

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