  Symmetric Positive Definite (SPD) matrices have been used in many fields of
medical data analysis. Many Riemannian metrics have been defined on this
manifold but the choice of the Riemannian structure lacks a set of principles
that could lead one to choose properly the metric. This drives us to introduce
the principle of balanced metrics that relate the affine-invariant metric with
the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric
with the Euclidean and log-Euclidean metrics. We introduce two new families of
balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics
and we discuss the relation between this new principle of balanced metrics and
the concept of dual connections in information geometry.
