Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive definite matrix whose non-diagonal elements are all non-positive, determine whether all the non-diagonal elements of $A^{-1}$ are non-negative or not.
This is one problem of my final exam on linear algebra, and I didn't solve it during the examination.
I verified the cases of $n \leq 3$ and guess that this should be correct, but I don't know how to deal with non-diagonal elements of a matrix properly.
Best Answer
Hints. Let the diagonal part and off-diagonal part of $A$ be $D$ and $-F$ respectively. Let $X= D^{-1/2}FD^{-1/2}$. Justify that $X\ge0$ entrywise, $\|X\|_2<1$ and $$ A^{-1}=D^{-1/2}(I-X)^{-1}D^{-1/2}=D^{-1/2}(I+X+X^2+\cdots)D^{-1/2}. $$ The rest should be easy.