The Reverse Order Law and the Riccati Equation

We give a full analytic solution to a particular case of the algebraic Riccati equation $XW{W}^{\ast }WX=W^{\ast }$ for any matrix $W$ (possibly non-square or non-symmetric) in using the Schur method, terms of the SVD decomposition of $W$. In particular, $(WX{)}^3=WX$ and $(XW{)}^3=XW$ for any solution $X$. We show that for $W=AB$, matrix $X=B^{+}{A}^{+}$ is a solution of this equation if and only if the reverse order law holds, i.e., ${(AB)}^{+}=B^{+}{A}^{+}$. For a Hermitian and invertible $W$ the maximal and stabilizing Hermitian solutions is shown to be equal to $W^{+}$. Equivalence to the equation $XWX=W^{+}$ is proven.