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 positive-definite. Furthermore, $J$ and $A$ are with full-row rank.



On solution of a recursion with rectangular matrices
If $J$ has more columns than rows, the map $S \to J S J^T$ is not one-to-one, so your equation does not determine $S(k+1)$.


