This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated.

Consider an $n \times n$ correlation matrix A such that all the off-diagonal entries are between [-1,0]. (**Note**: A correlation matrix is a positive semi-definite symmetric matrix, with diagonal entries 1 and all off-diagonal entries between [-1,1]).

Let $\alpha_i = \frac{\sum_{j=1,j \neq i}^{n}|A_{ij}|}{n-1}$ denote the mean of magnitudes of off-diagonal entries in $i^{th}$ column.

Let $v_{min} = [v_1,v_2,...,v_n]^T$ be the unit eigenvector corresponding to the least eigenvalue $\lambda_{min}$ of A. Let $v_k$ be the weight with minimum magnitude in $v_{min}$.

**Then empirically, I am observing that $\alpha_k$ is also minimum among all $\alpha_i$'s.**

I am wondering if this is indeed true and can be proved, or otherwise, if there is any counterexample where this will break?