I am trying to understand the proof of Theorem 4.3 of S. Friedland. Convex Spectral Functions. Linear Multilinear Algebra, 9:299--316, 1981.
The theorem states as follows: Let $A^{-1}$ be an M-matrix. Then the spectral radius $\rho(DA)$ of $DA$ is a convex functional on the space of $n\times n$ nonnegative diagonal matrices.
In the proof the authour mentioned that
$\frac{\partial \rho(DA)}{\partial d_{i}}|_{D^{0}}=\eta^{T}\frac{\partial \rho(D)}{\partial d_{i}}A\xi= \rho(D_{0}A)\frac{\eta_{i}\xi_{i}}{d_{i}^{0}} \quad i= 1, \dots n,$
where $D^{0}=\text{diag}\{d_{1}^{0}, \dots, d_{n}^{0}\}$ and all diagonal elements of $D^{0}$ positive; $\eta$ and $\xi$, respectively are the left and right eigenvectors of $D^{0}A$ corresponding to eigenvalue $\rho(D^{0}A)$ such that $\eta^{T}\xi =1$.
I do not understand how to obtain the last equality in this equation.
It will be helpful if someone can explain this to me.
Thank you
