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Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, $x_i>0$. I know that $\lambda_1\in (a_{ss}-r_s, a_{ss})$, where $a_{ss}=\min(a_{ii}, i=\overline{1,n})$, and $r_s=\sum_{j\neq s}|a_{sj}|$. My question is whether this apriori knowledge of the location of the smallest eigenvalue necessarily implies that $\max(x_i, i=\overline{1,n})=x_s$.

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