Commutators with maximal Frobenius norm

It is known that for any nonzero complex n×nn×n matrices XX and YY the quotient of Frobenius norms‖XY-YX‖F‖X‖F‖Y‖Fdoes not exceed 2. However, except for some special cases, only necessary conditions for attaining this bound have been found so far. We will completely characterize the pairs of matrices that satisfy equality with the quotient’s maximum.