Suppose we have a $n\times n$ block matrix
$$ X= \begin{bmatrix} A_{11} & \ldots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{n1} & \ldots & A_{nn} \end{bmatrix}, $$ where each block $A_{ij}$ is $d_i\times d_j$ and the diagonal blocks $A_{ii}$ are symmetric.
If there exists a block $A_{ii}$ such that $\rho(A_{ii})>1$, can we say anything about $\rho(X)$, for example, $\rho(X)>1$?
For instance, we could think of $X$ as a Jacobian matrix of a vector field that describes a $n$-player optimization dynamics. My conjecture is that the overall dynamics would be unstable, $\rho(X)>1$, if there is a player who is unstable under its own optimization dynamics, $\rho(A_{ii})>1$.
A quick search with math software gave me several counterexamples:
$$\begin{pmatrix}0. & -0.3 \\ 0.8 & 1.1\end{pmatrix} \mapsto \lambda=0.8, 0.3$$ $$\begin{pmatrix}1.2 & -1 & 0.3 \\ 0 & 0 & -0.3 \\ -1 & 0.5 & -0.5\end{pmatrix} \mapsto \lambda=0.643, 0.461, -0.404$$
One can imagine replacing each number by a block matrix of the same order of magnitude (e.g. $a\mapsto aI$) and the results should still hold.
In the context of dynamics, one variable -- the prey -- may increase exponentially if isolated, but the other variables -- the predators -- may restrict it from growing.