upper bound on the difference between two Perron-Frobenius eigenvalues Let $\lambda, \mu$ be the Perron-Frobenius eigenvalues of two non-negative matrices $A,B$ respectively. I am interested in knowing whether there are any results available on the upper bound of $|\lambda-\mu|$ in terms of norms of $\|A-B\|$.
 A: I guess you want a bound in terms of a norm of $A-B$, since one obviously has $|\rho(B) - \rho(A)| \leq ||A|| + ||B||$ for any operator norm $|| \cdot ||$. 
If $A$ is irreducible, then its Perron eigenvector $x_A$ (normalized so that $||x_A||_1 = 1$) has strictly positive entries, hence $K(A) = \max_i (x_A)_i^{-1} < \infty$. In that case one has
$$
(*)   \ \ \ \ \ \ \ \ \|\rho(B) - \rho(A)| \leq K(A) ||B - A||_{\infty}, 
$$
where $||C||_{\infty} = \max_{i,j} |c_{ij}|$. Indeed,
$$
(\rho(B) - \rho(A)) \langle x_{A} | x_{B^*}  \rangle = \langle x_{A} | B^* x_{B^*}  \rangle  - \langle A x_{A} | x_{B^*}  \rangle  = \langle (B-A) x_{A} | x_{B^*}  \rangle,
$$
and the conclusion follows from the inequalities 
$$
\langle x_{A} | x_{B^*}  \rangle \geq K(A)^{-1} ||x_{B^*}||_1 = K(A)^{-1}
$$
and
$$
|\langle (B-A) x_{A} | x_{B^*}  \rangle | \leq || B-A||_{\infty} ||x_A||_1  ||x_{B^*}||_1 = || B-A||_{\infty} .
$$
Note that $(*)$ is still valid for general $A$, if one replaces $K(A)$ by $\max(K(A),K(B)$, with $K(A)= \max_{j} K(A_j)$, where $(A_j)_j$ are the irreducible blocks of $A$.
