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Eigenvector of a nonnegative matrix in closed form

If you write $A\nu=\nu$ as a system of equations it can be written as $$ \frac{\nu_i}{1-\alpha_i}=\sum_{j=1}^{n}\frac{\alpha_i\nu_j}{1-\nu_j} \qquad \forall i=1, ..., n. $$ Then the change of variabl …
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