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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.

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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.

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