# Approximating a matrix by commuting matrices

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.

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.