For a linear system, $x(t+1)=Ax(t)$, we know the condition for Lyapunov condition is $$A^{\top}PA-P<0.$$ This comes directly from the Lyapunov function $V(x)=x^{\top}Px$. Somehow, this condition has an equivalent one called 'observer form', which is $$AQA^{\top}-Q<0.$$ I wonder how this form is derived. Either from the system itself or from any linear algebraic transformation.
Extension: My original problem considers a switching system. Suppose we have a set of $A_i$, $i\in\{1,\cdots,n\}$. Further suppose there exists a common $P$, such that for all $i$, $$A_i^{\top}PA_i-P<0.$$ (Due to the existence of a common PD $P$, no matter how the system switches, it is stable.) Then is it equivalent to say: there exists a common PD $Q$, such that for all $i$, $$A_iQA_i^{\top}-Q<0.$$
Thanks.
Ok, I found the answer. It can be simply proved by Schur complement and LMI.
Since $P$ is PD, $$A_i^{\top}PA_i-P<0 ~\Leftrightarrow~ \begin{bmatrix}P^{-1}& A_i\\A_i^{\top} & P \end{bmatrix}>0$$ Now, let $Q=P^{-1}$, one has $$\begin{bmatrix}Q& A_i\\A_i^{\top} & Q^{-1} \end{bmatrix}>0 ~\Leftrightarrow~ A_iQA_i^{\top}-Q<0 $$