The Reverse Order Law and the Riccati Equation

TL;DR

This paper provides an explicit analytic solution to a specific algebraic Riccati equation using the SVD, explores conditions for the reverse order law, and characterizes solutions for certain matrix classes.

Contribution

It offers a full analytic solution to a Riccati equation involving the SVD and characterizes solutions related to the reverse order law for matrices.

Findings

Solution expressed via SVD decomposition.

Equivalence of solutions to the reverse order law.

Maximal Hermitian solution equals the Moore-Penrose inverse for invertible W.

Abstract

We give a full analytic solution to a particular case of the algebraic Riccati equation $XWW^*WX=W^*$ 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.