Surjective isometries are likely the most important "symmetries" of metric spaces. When

the underlying space is also equipped with some compatible algebraic structure, it frequently turns out that the (surjective) isometries are closely related to corresponding algebraic isomorphisms.

In this talk we are concerned with that phenomenon in areas of linear algebra and functional

analysis.

In the first part of the talk we present some classical results on linear isometries of matrix algebras. In the second part we consider non-linear problems on the positive definite cone in matrix algebras and explain how certain algebraic considerations

help to characterize their isometries with respect to certain important metrics. If time permits, we will comment on infinite dimensional extensions, too.