I am trying to prove that the inverse of a $\mathbf{positive}, \mathbf{symmetric}, \mathbf{diagonally-dominant}, \mathbf{doubly-stochastic}$ matrix and having the following $\mathbf{property (3)}$ - has positive diagonal elements and negative non-diagonal elements.
I'll describe the below definitions.
1)Diagonally dominant matrix - $|a_{ii}| > \sum_{i \neq j}|a_{ij}| \quad \forall i$
2) Doubly stochastic matrix - sum of entries in any row or column is unity. i.e., $\quad \sum_{i} a_{ij} = \sum_{j} a_{ij} = 1 \quad \forall i,j$
3) Let $b_1, b_2,..,b_{N-1}$ be a decreasing sequence of positive numbers. The non-diagonal elements of the matrix satisy: $$\qquad a_{ij} = b_{|i-j|} \qquad \forall i \neq j$$
Need to show that the inverse of such a matrix $\mathbf{A}$ will have positive diagonal and negative non-diagonal elements.
I have come till proving that the row sum and column sum of $\mathbf{A}^{-1}$ is 1. But not able to prove further.
That was an intriguing conjecture, but unfortunately it is not always true.
Consider a 5x5 matrix $A$ of the form you described with $b_1=0.15$, $b_2=b_3=0.05$ and $b_4=0.0125$.
The inverse of this matrix has several strictly positive off-diagonal elements ($a^{-1}_{15}=a^{-1}_{51}\approx0.0189$ and $a^{-1}_{24}=a^{-1}_{42}\approx0.004$).