2024/08/12 by Oskar Kędzierski, Kędzierski, Oskar
Engineering · Mathematics · #15A010 #15A09 #93B40 #FOS: Mathematics #Material Science and Thermodynamics #Modeling, Simulation, and Optimization #Optimization and Control (math.OC) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.09035
openalex publication_date 2024/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.