I have a (column) sub-stochastic matrix, $P$, which is nonnegative and irreducible (hence satisfies the criteria of the Perron Frobenius theorem). I know its dominant eigenvalue, $\rho$ and left and right eigenvectors corresponding to $P$. That is, $P v = \rho v$ and $u^T P = \rho u^T$. Additionally I use the following normalizations: $\sum_i v_i = 1$ and $\sum_i u_i v_i = 1$.
Now, I perturb a single column of $P$ by a constant, encoded in matrix $E$. Under consideration is the new eigenvalue equation: $(P+E)v' = \rho' v'$. (I assume that the perturbation is not too large so that $P+E$ still satisfies P.F. thm). Expanding, and writing $\rho' = \rho + \delta \rho$ and $v'=v+\delta v$;
\begin{equation} Pv + P \delta v + Ev + E \delta v = \rho v + \rho \delta v + \delta \rho v + \delta \rho \delta v \end{equation}
Removing second order terms, and using the original eigenvector equation, we have \begin{equation} P \delta v + Ev= \rho \delta v + \delta \rho v \end{equation} Dotting with $u^T$ on both sides and using normalization tells us \begin{equation} u^T P \delta v + u^T E v = \rho u^T \delta v + \delta \rho u^T v \end{equation}
or \begin{equation} u^T E v = \delta \rho u^T v \end{equation} which is great, as we now have a first order estimate of the perturbation of the Perron root; namely $\delta \rho = u^T E v.$
However, I am also interested in the first order perturbation to the entries of left and right eigenvectors $u, v$.
I do not know the other non-dominant eigenvectors (so I do not want to use them as a basis), but I am guessing we can return to the vector equation:
\begin{equation} P \delta v + Ev= \rho \delta v + \delta \rho v \end{equation} which can be rewritten as $(P - \rho \mathbf{1}) \delta v = (\delta \rho \mathbf{1}-E)v$, where the RHS is now a calculable vector.
How can I extract a first order perturbation for the entries of $\delta v$?
Extra things to keep in mind; $\sum_i \delta v_i = 0$, $u$ is normal to $\delta v$ in first order. Also, I do not want to attempt to invert $P-\rho \mathbf{1}$, as it is usually non-invertible. I also do not want to use the generalized matrix inverse. Computations involving either of these are usually more difficult than simply calculating the new Perron vectors after all!
TIA for any help
P.S. - I am also especially interested in (simple) results for the derivative of $u$ and $v$ w.r.t. the matrix entries $P$. I have seen some results by Elsner, Neumann, Lu, etc. But most deal with the generalized matrix inverse.