TIME INVARIANCE
Let " $ \Rightarrow $ " represent a system.
If for any signal $ X(t)\Rightarrow Y(t) $ implies that $ X(t - t_o)\Rightarrow Y(t - t_o) $ then the system is time invariant.
TIME-INVARIANT SYSTEM
$ X(t)\Rightarrow Y(t) = a*X(t) $ where $ a \in \mathbb{{C}} $ is a time invariant system.
PROOF
$ X(t)\Rightarrow Y(t) = a*X(t) \to [time delay] \to Z(t) = Y(t - t_o) = a*X(t - t_o) $
$ X(t)\to [time delay] \to Y(t) = X(t - t_o) \Rightarrow W(t) = a*Y(t) = a*X(t - t_o) $
$ W(t) = Z(t) \Rightarrow $ The system is time-invariant
