How to solve the matrix equation XA-AX=f(X)

2010/05/18 by Gérald Bourgeois, Bourgeois, Gerald
Computer Science · Mathematics · #15A24 #Advanced Optimization Algorithms Research #Algebraic and Geometric Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1005.3159

openalex publication_date 2010/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be an analytic function defined on a complex domain Omega and A be a (n,n) complex matrix. We assume that there exists a unique alpha satisfying f(alpha)=0. When f'(alpha)=0 and A is non derogatory, we solve completely the equation XA-AX=f(X). This generalizes Burde's results. When f'(alpha) is not zero, we give a method to solve completely the equation XA-AX=f(X): we reduce the problem to solve a sequence of Sylvester equations. Solutions of the equation f(XA-AX)=X are also given in particular cases.

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