Properties of a Kalman filter of a non controllable system Let's say I have a standard system 
\begin{align}
x(t+1)&amp;=Ax\\
y(t) &amp;=Cx(t) 
\end{align}

As you can see $B=0$, so the system is not controllable. 

For the steady state Kalman filter I'd say that if $A$ is Hurwitz then it converges. 

But what if $A$ is say,  $\mathbf I_n$ (i.e. eye(n)), what can I say about the optimality of the SS.  filter varying $C$?
In this setting you are considering an observation problem. Controllability does not play a role. In geenral, the condition under which you can obtain a convergent estimate is that the couple $(A,C)$ is detectable, i.e. that you can design a gain matrix $L$ so that $A+LC$ is Hurwitz. Under this hypothesis the eigenvalues of $A+LC$ will be with negative real-part and they are assigned in a optimal way, in the kalman sense.
The kalman gains depend on $C$, therefore as you vary $C$ you should vary the gains accordingly to obtain optimality