Review Note

Soit \(M \in \mathrm M_n(\mathbb R)\). On pose \(\begin{array}{} \varphi_M : & \mathrm M_n (\mathbb R) & \to & \mathrm M_n(\mathbb R) \\ & A & \mapsto & AM\end{array}\). Montrer que \(\varphi_M\) est diagonalisable si et seulement si \(M\) l'est.

• {{c1::On a pour tout \(P \in \mathbb R[X]\), \(P(\varphi_M) = \varphi_{P(M)}\).}}
• \(\Rightarrow\) {{c1::\(\pi_{\varphi_M}(\varphi_M) = \varphi_{\pi_{\varphi_M}(M)} = 0\) donc \(\pi_{\varphi_M}(M) = \varphi_{\pi_{\varphi_M}(M)}(I_n) = 0\) et \(\pi_M | \pi_{\varphi_M}\).}}
• \(\Leftarrow\) {{c1::\(\pi_M(\varphi_M) = \varphi_{\pi_M(M)} = 0\) donc \(\pi_{\varphi_M} | \pi_M\).}}