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Let $ A = (a_{ij})$ be a matrix of size $ n^2 $ such that $ a_{ij} > 0$. Let $x = (x_1,\ldots,x_n ) $ and$ y = (y_1,\ldots,y_n) = Ax$. Show that there exist an eigenvalue $\lambda $ of A, such that: $$ \lambda \in \left[ {\min \left\{ {\frac{{y_i }} {{x_i }}} \right\}_{i = 1}^n ,\max \left\{ {\frac{{y_i }} {{x_i }}} \right\}_{i = 1}^n } \right]. $$ I have no idea how to do it :S!!

Someone knows this result or related? or some solution?

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    Nothing special about $\mathbf A$?2011-11-19
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    You may want to look into the Perron-Frobenius theorem. I believe this problem is an exercise in Meyer's "Matrix Analysis."2011-11-19

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