Hello,

I have the following problem:

Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), where $A$ is a given matrix and $y$ is a known vector. Also, I would like the rows of $L$ to sum to one, but perhaps that condition can be dropped.

Is there a known way of approaching this problem, or is it NP-complete? Basically the issue is that I need both $L$ and $L^{-1}$ to satisfy a certain property. Surprisingly, I haven't found any results on optimization problems that involve a matrix and its inverse.

Thanks a lot!