I want to prove the positive semi-definiteness of a $N\times N$ matrix A, where $a_{ij}=1-\rho\min\{|i-j|,N-|i-j|\}$, $|i-j|$ is the absolute difference between i and j, $\rho$ is a positive real number and all $a_{ij}$ are nonnegative. Then all diagonal elements of A are 1. By Perron-Frobenius theorem I can show A's largest eigen-value (which is also A's spectral radius)is positive, but I have no idea to prove the positiveness of other eigen-values.
One way I tried before is to normalize A, since the sum of each of its rows and columns keeps the same and all A‘s elements are non-negative, therefore A could be a symmetric doubly stochastic matrix after normalization. While I still didn't find any way to solve it and all my simulation results show the eigen-values of A are non-negative. Any help? thank you so much.