Matrices with extremal commutants and beyond

Matrices whose commutant is either maximal or minimal with respect to set-inclusion were classified in 2005 by Dolinar and Šemrl in their pursuit towards classification of bijections which preserve zeros of Lie product. The classification which they obtained  is valid only for complex matrices. Recently, we extended their classification in several directions:
(a) For matrices over an arbitrary field. Besides the  classes that were obtained in the complex case some new possibilities are possible here.
(b) We also investigated problems of these kinds within certain subclasses of (complex) matrices, in particular within doubly stochastic matrices.
(c) Instead of commutativity one may further consider the commutativity up to a factor \xi, defined by AB = \xi BA and classify matrices which are extremal with respect to this relation.

The aim of the talk is to present the obtained results. This is joint work with many collaborators.