Dear all,

As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}\(A+A^{T}\)$ for a stochastic $A$ (as a function of the singular values of $A$), which is generally not Hermitian. 

I am aware of the upper bounds (due to Weyl and Fan) and of the fact that for general $\sigma_{i}\(A+B\)$ no lower bound is known. Do you see a way?

Thank you.

Edit: A can be considered a power of a lazy row stochastic matrix. I.e., $A=P^k$ for some strongly diagonally dominant row stochastic $P$.