Symmetries of structures of positive (semi)definite matrices

In this talk we survey the results of our recent research on the ‘symmetries’ of different structures of positive (semi)definite matrices. In those results we give exact descriptions of certain algebraic endomorphisms/automorphisms (such as Jordan triple endomorphisms, sequential endomorphisms), isometries corresponding to different distances, and transformations preserving divergences (among others, different sorts of relative entropies).