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

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

So far, I am observing empirically that $v_k \leq \frac{1}{\sqrt(n)}$.

I am wondering if this is indeed true in general, or otherwise, if there is any counterexample where this will break?

**Note**: For the cases with algebraic multiplicities (e.g. identity matrix), since the set of eigenvectors are not unique for such matrices and hence technically one might choose an appropriate eigenvector that would satisfy the above observation.