Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ being the largest eigenvalue of $A(t)$ and $t$ denotes the time. If $A(t)$ changes over time with each time a random element in $A(t)$ is being increased by a random value, will the spectral radius be monotonously increasing over time as well?


In other words, can I say that:


$\rho(A(1))\leq\rho(A(2))\ldots\leq\rho(A(N))$


If yes, how do I proof this? 


If no, why?



Spectral radius of a time-varying matrix with strictly positive increment of the matrix's entry
Anthony and David are referring to the Perron-Frobenius Theorem, which easily proves the monotonicity you ask about. There is a precise quantitative result in my paper *Perturbations of Shifts of Finite Type*, SIAM J. Disc. Math. 2 (1989), 350-365:


Lemma 6: Let $A = [a\_{i,j}]$ be a real square matrix with simple eigenvalue $\rho(A)$ and corresponding (row) left eigenvector $v>0$ and (column) right eigenvector $w>0$. Then the matrix of partial derivatives of $\rho(A)$ with respect to the entries of $A$ is given by
$$
 \left[\frac{\partial \rho(A)}{\partial a\_{ij}}\right]=\frac{wv}{vw}.
$$


The proof is a simple application of the chain rule applied to $Aw=\lambda w$, where $\lambda=\rho(A)$, and is actually true for every simple (i.e. multiplicity one) eigenvalue of $A$. 


