Does any body know a reference on lower triangular matrices with negative entries everywhere except for the diagonal and subdiagonal where entries are positive (when all entries are negative with exception of the diagonal these matrices are called M-matrices) . The inverse of M matrices are nice since they consist of only positive numbers whereas the inverse "Quasi-M matrices" have alternating signs.
I am interested in the equation
$$ A_n x = b_n$$
Where $A^{i,i-1}_n - A_n^{i,i} \sim O(1/n)$ and $A_n^{i.i} \sim O(1)$, it seems natural to replace $A_n^{i,i-1}$ by $0$ and $A_n^{i,i}$ by $2 A_n^{i,i}$ to obtain an M matrix however I am not sure the solution of these two problems has to necessarily be near and it seems to depend on the value of $A_n^{-1} b_n$.
