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If $ A_{n \times n} $ is a transition matrix $ - $ a positive matrix in which the sum of all entries in each row or columns equals $ 1 - $ is it true that $ A $ must always have $ n $ linearly independent eigenvectors?

I know that $ 1 $ is an eigenvalue of $ A $ with multiplicity $ 1, $ but how about the other eigenvalues and their associated eigenvectors? Can we prove that all $ n $ eigenvectors are linearly independent?

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  • $\begingroup$ I would begin by considering possible $0,1$-matrices, just to keep things simple. $\endgroup$ Commented Jul 14, 2016 at 23:57

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Try

$$ \pmatrix{1/2 & 1/3 & 1/6\cr 0 & 1/3 & 2/3\cr 1/2 & 1/3 & 1/6\cr}$$

Or, if you want the entries strictly positive,

$$ \pmatrix{s & 1/3 & 2/3 - s\cr 1-2s & 1/3 & 2s - 1/3\cr s & 1/3 & 2/3-s\cr}\ \text{for} \ \frac{1}{6} < s < \frac{1}{2},\ s \ne \frac{1}{3}$$

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