The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as $$ \operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\Vert AX-XB\right\Vert_F}{\left\Vert X\right\Vert_F} $$ (see e.g. Golub and Van Loan, Section 7.2.4).
I would like to express the separation between two M-matrices in terms of their Perron vector and values $Au=\lambda u$, $Bv=\mu v$. All I can do is estimating $\operatorname{sep}(A,B)\geq \lambda+\mu$. Is there any better bound, maybe from above? The question looks like an "eigenvalues vs. singular values" bound, so I am not sure that the answer is positive.
Alternatively, do you know any "handier" way to deal with $\operatorname{sep}(A,B)$, other than using its definition and its alternative formulation as the smallest singular value of a Kronecker sum $\sigma_{min}(B \otimes I + I \otimes A^T)$? I always find it unwieldy to use this definition, it is not easy to squeeze something simple to compute/estimate out of it. For instance, if $A$, $A'$, $B$ are M-matrices and $A'\geq A$, does $\operatorname{sep}(A',B)\geq \operatorname{sep}(A,B)$ hold?
