I am trying to figure out the structure of M matrix whose inverse has a special form: Let $A$ be an inverse M matrix (inverse M matrices are those matrices whose inverse is an M matrix, https://en.wikipedia.org/wiki/M-matrix) such that each row sum of the matrix is a fixed constant (greater than 1). Each diagonal entry is strictly greater than all off-diagonal entries of that row such that $xa_{ii}\leq a_{ij}\leq ya_{ii} \forall i\neq j$, where $0<x<y<1$. I am trying to show that $A^{-1}$ will have positive diagonal entries less than 1.
So far, I could not find any reference on why such a result must hold true, but I could not get a counter-example either (I tried numerical examples). Intuitively, I think it will be true due to the special structure of the matrix, the off-diagonal entries of $A$ are close to each other, so the inverse will have the above structure to compensate for that. Any idea or suggestion will be really helpful.
Inverse M matrices structure
linear algebramatrices
Best Answer
I got 2 counter-examples:
Here $x=0.8, y=0.9$, so, $(0.8)(1.1)\leq 0.89\leq (0.9)(1.1)$.
Here $x=0.94, y=0.98$, so, $(0.94)(5.1)\leq 4.9\leq (0.98)(5.1)$.
So, I guess intuition does not always work in research.