A strange phenomenon for the singular values of commutators with rank one matrices

Abstract

The singular values of XY ô Y X are the objects under investigation. Here, X andY are square matrices with complex entries, and one of them has rank one. Hence, there are at most two non-trivial numbers among the commutator's singular values, and the pairs of interest can be depicted in the plane. The emphasis will lie on the unexpectedly intriguing case in which both matrices are of rank one { because the result then is astonishingly complex, and the problem gives rise to interpretations unveiling geometry acting in the background.