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

TL;DR

This paper provides a comprehensive method for solving the matrix equation XA-AX=f(X) for analytic functions f, generalizing previous results and offering solutions based on matrix properties and Sylvester equations.

Contribution

It introduces new complete solutions for the matrix equation XA-AX=f(X) under various conditions, extending prior work by Burde.

Findings

Complete solutions when f'(alpha)=0 and A is non derogatory

Reduction to Sylvester equations when f'(alpha)≠0

Solutions for f(XA-AX)=X in specific cases

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.