Approximating a matrix by commuting matrices

Question

Let $A$ be an $n\times n$ matrix with non-negative entries. Can we approximate $A$ by matrices with strictly positive entries that commute with $A$? This question came up in connection with the s-called $Q$-matrices in Markov Chain Theory. Thanks in advance for any ideas.

Answer 1

Clearly this is not always possible. E.g. if $A$ is a positive diagonal matrix with distinct diagonal entries, it only commutes with diagonal matrices.