The maximum eigenvalue of the sum of two matrix

Question

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 $C$.

If $x$ is a common eigenvector of $A$ and $B$, then $\lambda\leq \lambda_1,\ \lambda\leq \lambda_2$. — Samrat Mukhopadhyay, Jul 17 '13 at 13:05
@SamratMukhopadhyay I think you meant $$\lambda x=Cx=Ax+Bx\leq\lambda_1x+\lambda_2x,$$ and hence $\lambda\leq\lambda_1+\lambda_2$. — AD - Stop Putin -, Jul 17 '13 at 14:02
@AD Yes, yes, I wrote it incorrectly, thinking something else. — Samrat Mukhopadhyay, Jul 17 '13 at 14:05

score 3 · Accepted Answer · answered Jul 17 '13 at 16:05

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.

The maximum eigenvalue of the sum of two matrix

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