Approximating commuting matrices by commuting diagonalizable matrices
Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that
Moreover, it would be nice if the following property was additionally satisfied: if $A,B$ are real, then $A_n,B_n$ can be chosen to be real as well.
P.S. I asked this question on math.SE a couple of days ago.