Abstract
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 on the spectrum of the matrices that achieve the equality in those inequalities. As an application, we complete some previous results concerning Procrustes problems for unitarily invariant norms in the manifold of positive definite matrices.
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