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We all know that the spectral radius of a stochastic matrix is $1$. But how's the "negative" proposition: For a matrix $M=(m_{ij})_{n\times n}$, if there exists $m_{ij}<0$ for some $1\le i,j\le n$ and $Me=e$ where $e$ is a vector with all entries being $1$, the spectral radius $\rho(M)>1$. Thank you.

UPDATE: the conditions seem to be too general so the answer turns to be trivial. I've posted another question at About the spectral radius of a kind of matrices which is closer to the actual problem I've met. Thank you.

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Try the matrix: $ \left( \begin{array}{cc} 2 & -1 \\ 2 & -1 \end{array} \right)$. It has two eigenvectors, $(1,1)$ and $(1,2)$, with respective eigenvalues $1$ and $0$.

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  • $\begingroup$ Hmm...the problem I've actually met seems to have some implied constraints. Anyway thanks for the answer. $\endgroup$
    – ziyuang
    Apr 17 '13 at 0:47
  • $\begingroup$ I've updated the question at math.stackexchange.com/q/364427/11014 $\endgroup$
    – ziyuang
    Apr 17 '13 at 13:33

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