Let $V = \mathbb{M}_n(\mathbb{R})$, the vector space of $n\times n$ matrices.
Define the linear transformation $\alpha: V \to L(V,V)$ that leads a matrix $M \in V$ to $\alpha(M): P \mapsto MP-PM$.
I want to proof that, if $M$ has $\lambda_1,...,\lambda_n$ different eigenvalues, then $\alpha(M)$ will have eigenvalues $\lambda_i-\lambda_j$, with $1 \leq i < j \leq n$.
I know that $MP$ and $PM$ has the same eigenvalues, and I was thinking of diagonalize M to try to get somewhere, but I didn't find anything. Any hints?
