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arXiv · 2409.09035

The Reverse Order Law and the Riccati Equation

Oskar Kędzierski
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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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Oskar Kędzierski. 2024-08-12. The Reverse Order Law and the Riccati Equation. https://arxiv.org/abs/2409.09035

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