  Given two positive definite matrices $A$ and $B$, a well known result by
Gelfand, Naimark and Lidskii establishes a relationship between the eigenvalues
of $A$ and $B$ and those of $AB$ by means of majorization inequalities. In this
work we make a local study focused in the spectrum of the matrices that achieve
the equality in those inequalities. As an application, we complete some
previous results concerning Procustes problems for unitarily invariant norms in
the manifold of positive definite matrices.
