Quasi-stochastic
In order not to make the title too long I used the term Quasi-Stochastic
with this meaning: a quasi-stochastic matrix $Q$ is a square matrix $Q = (q_{i,j}) \in \mathbb{R}^{n \times n}$ where all entries are bound to the interval $q_{i,j} \in ]0,1[$ and where all rows sum to a number lower than 1 (strictly):
$\sum_{j=1}^n q_{i,j} < 1$
This condition determines what I call the quasi-stochastic
thing.
The question: specific case
Consider a quasi-stochastic matrix $Q$ and its eigenvalues $\lambda_i$. I would like to understand if the following equation holds:
$ |\lambda_i| < 1, \forall i = 1 \dots n $
Or, less strictly
$ |\lambda_i| \leq 1, \forall i = 1 \dots n $
Rationale
There is a reason why I ask. If you consider a Markov chain and its transition probabilities matrix $P$, if the chain is ergodic than the matrix has all its eigenvalues in the unit circle with one eigenvalue on the edge of it. If create a reduced version of this matrix, what happens to the eigenvalues? Using Matlab I could try some experiments and experienced that all eigenvalues are in the circle and the eigenvalue which is $\lambda_1 = 1$ falls to $\lambda_1 < 1$. However how to get mathematical evidence?