The matrix equation $XA-AX=f(X)$ when $A$ is diagonalizable

TL;DR

This paper investigates the solutions to the matrix equation XA - AX = f(X) for diagonalizable matrices A over an algebraically closed field, extending understanding of such equations with polynomial or holomorphic functions.

Contribution

It characterizes all solutions of the matrix equation when A is diagonalizable, for polynomial or holomorphic functions f, broadening the theoretical framework of matrix equations.

Findings

Complete solution set characterization for diagonalizable A

Extension to polynomial and holomorphic functions f

Insights into the structure of solutions for complex matrix equations

Abstract

$K$ is an algebraically closed field with characteristic $0$ and $f$ is a polynomial or a holomorphic function. We study all solutions of the equation $XA-AX=f(X)$, in the unknown $X\in M_n(K)$, when $A\in M_n(K)$ is diagonalizable.