This is about general equilibrium:
Suppose that $x(t)$ represents outputs of all sectors and parts of the whole economy - represented as matrix. How outputs evolve to $x(t+1)$ is determined by the matrix $A$ - $A$ representes how previous outputs are used as inputs to produce new ouputs - so $A$ can be said as table of outputs produced from inputs. $$x(t+1) = A \cdot x(t)$$
and it continues on to say that if the matrix $A$ has (absolute value of) all eigenvalues less than 1, the system is stable, while if not, is unstable.
The question is, I do get that when when $x$ is the eigenvector of $A$, it becomes unstable, but what if it's not - then can we still say that the system is unstable? (so what I am saying is, let's assume that the system will never have $x$ as eigenvector of $A$. Then what happens?) If so, can anyone show the proof of it?
(The linked question seems to assume that $x$ is an eigenvector of $A$ - I do not assume that at here.)