TITLE: The maximum eigenvalue of the sum of two matrix
QUESTION [9 upvotes]: Suppose there are two matrix $A$ and $B$. The components of each matrix is non-negative.
And 
$$Ax_1=\lambda_1 x_1  $$ where $\lambda_1$ is the maximum eigenvalue of $A$.
Similarly
$$Bx_2=\lambda_2 x_2  $$ where $\lambda_2$ is the maximum eigenvalue of $B$.
Let $C = A+B$
And 
$$Cx=\lambda x  $$ where $\lambda$ is the maximum eigenvalue of $C$.
From wiki(https://en.wikipedia.org/wiki/Matrix_norm), it shows that $$\left\| A+B \right\|\le \left\| A \right\|+\left\| B \right\|$$
The maximum eigenvalue is 2 norm. So no matter the components are negative or not, $$\lambda\le \lambda_1+\lambda_2$$

REPLY [3 votes]: Yes. By Gelfand's formula, if $\|\cdot\|$ is any matrix norm, then $\rho(A)=\lim_{n\to\infty}\|A^n\|^{1/n}$. Now, simply take $\|\cdot\|$ as the maximum row sum norm, i.e. $\|A\|=\max\limits_i\sum_j|a_{ij}|$. Since both $A$ and $B$ are nonnegative, we have $0\le A^n\le(A+B)^n$ entrywise. Hence the result follows.