I faced to the following discrete
$$ y[n+1]=Ay[n]+Bx[n]+\eta[n] $$
where $y[n] \in \mathbb{R}^n$, $A$ and $B$ are matrices with appropriate dimension, and $\eta[n]$ is noise.
I have no problem with the following
$$ y[n+1]=Ay[n]+Bx[n] $$
Because I can write the z-transform as follows
$$ zY(z)-zY(0)=AY(z)+BX(z) $$
Can I simply write the following even for noise?
$$ zY(z)-zy(0)=AY(z)+BX(z)+ H(z) $$
Please answer it conceptually and explain it.