Suppose I have a positive semi-definite matrix A and a vector x with positive entries. I want to establish upper and lower bounds on the entries of the vector $(I+A)^{-1}x$. Using the $p_\infty$-norm I can get upper bounds on the absolute size of entries of $x$ using
$$\|(I+A)^{-1}x\|=\left(\min_{y≠0}\frac{\|(I+A)y\|}{\|y\|}\right)^{-1}\|x\|$$
However, I also need to get lower bounds for their absolute size. Is there a way to do this?
Finally, a question on signs: if $x$ is the 1-vector, is $(I+A)^{-1}1$ necessarily positive or can it be negative?
UPDATE: The answer to the last question is no.
Set $A= \begin{bmatrix} 10 & 3 \\ 3 & 2 \end{bmatrix} $. Then $(I+A)^{-1}=\frac{1}{13} \begin{bmatrix} 2 & -3 \\ -3 & 11 \end{bmatrix} $ whose first row sum is negative.