I was wondering if there is a way to get some conditions on the main diagonal elements of a particular matrix. I'm interested in the sign of the elements. The thing is that, I don't want to compute the diagonal elements one by one, but rather use (if any) a theorem that (under certain assumptions) helps me to gain some insight on the sign of the main diagonal entries.
Consider $A$, Hurwitz, Companion and Invertible (and then $A$ has only negative or zero elements on the main diagonal). Is it possible to say anything about the diagonal elements of: $M=(A+A^{-1})^{-1}$?
And of course, without computing the elements of $M$ explicitly.
Thanks in advance!!!!