Greetings to members here. The question is how to calculate the solution $S(k)$ of the following recursive equation $$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$ where $J$ and $A$ are rectangular not square. $R$ is positivedefinite. Furthermore, $J$ and $A$ are with fullrow rank.

If $J$ has more columns than rows, the map $S \to J S J^T$ is not onetoone, so your equation does not determine $S(k+1)$. 

