The Reverse Order Law and the Riccati Equation

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Détails bibliographiques
Auteur principal:	Kędzierski, Oskar
Format:	Preprint
Publié:	2024
Sujets:	Rings and Algebras Optimization and Control 15A09, 15A010, 93B40
Accès en ligne:	Acceder al recurso
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_version_	1866917775489892352
author	Kędzierski, Oskar
author_facet	Kędzierski, Oskar
contents	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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2409_09035
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	The Reverse Order Law and the Riccati Equation Kędzierski, Oskar Rings and Algebras Optimization and Control 15A09, 15A010, 93B40 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.
title	The Reverse Order Law and the Riccati Equation
topic	Rings and Algebras Optimization and Control 15A09, 15A010, 93B40
url	https://arxiv.org/abs/2409.09035

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