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Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each row is 1). I wish to prove that all the eigenvalues of $M$ are real. Do you think this is true?

user67490
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No. The eigenvalues of $ \begin{pmatrix} 1/2 & 1/3 & 1/6 \\ 1/6 & 1/2 & 1/3 \\ 1/3 & 1/6 & 1/2 \\ \end{pmatrix}$ are $1,\ 1/4+i\sqrt{3}/12,\ \text{and } 1/4-i\sqrt{3}/12.$

  • I was looking for a $2\times 2$ counterexample...to finally come to the conclusion that the fact is true in this case. So your counterexample is the best possible, +1. – Julien Apr 18 '13 at 12:51
  • The eigenvalues for $2\times 2$ stochastic matrices are both real. I had explored the $3\times 3$ case earlier here: http://math.stackexchange.com/questions/141212/eigenvalues-for-3-times-3-stochastic-matrices –  Apr 18 '13 at 12:54
  • Yes, I figured this out when looking for a counterexample. Thanks for the link. – Julien Apr 18 '13 at 13:01
  • No problem. Thanks for the upvotes! –  Apr 18 '13 at 13:02
  • You're most welcome: it is always a pleasure to upvote nice answers/questions. – Julien Apr 18 '13 at 13:05