$GL_n(\mathbb{F})$ is a linear space consists all $n\times n$ matrices over filed $\mathbb{F}$. Suppose $A$ is semisimple, prove that the linear transform $$ f_A:\;X\to XA-AX,\;\forall X\in GL_n(\mathbb F) $$ is also semisimple.
$A$ is semisimple, so there exists an invertible matrix $P$, $\Lambda=PAP^{-1}$ is a diagonal matrix. So $f_{PAP^{-1}}=X\Lambda-\Lambda X$. What is the relation between $f_A$ and $f_\Lambda$, and how to find a base that the matrix representation of $f_A$ is diagonal.
