# The Reverse Order Law and the Riccati Equation

Source: [The Reverse Order Law and the Riccati Equation](<https://www.sota2.com/research/paper/68fa5c4a1e209131ce9cfa75>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T17:32:29.107Z

Authors: Oskar K\\k{e}dzierski

Published: 2024-08-12T07:50:32.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2409.09035>)

[PDF](<https://arxiv.org/pdf/2409.09035.pdf>)