Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. Let $H$ be an element of $\mathcal { E }$. a) Calculate the image under $\Phi _ { H }$ of the canonical basis of $\mathcal { M } ( n , \mathbb { K } )$. Deduce that $\Phi _ { H }$ is a diagonalisable endomorphism of $\mathcal { M } ( n , \mathbb { K } )$. b) Show that there exists a basis of $\mathcal { A }$ in which the matrices of the endomorphisms of $\mathcal { A }$ induced by the $\Phi _ { H }$, for $H \in \mathcal { E }$, are diagonal.
Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. Let $H$ be an element of $\mathcal { E }$.
a) Calculate the image under $\Phi _ { H }$ of the canonical basis of $\mathcal { M } ( n , \mathbb { K } )$. Deduce that $\Phi _ { H }$ is a diagonalisable endomorphism of $\mathcal { M } ( n , \mathbb { K } )$.
b) Show that there exists a basis of $\mathcal { A }$ in which the matrices of the endomorphisms of $\mathcal { A }$ induced by the $\Phi _ { H }$, for $H \in \mathcal { E }$, are diagonal.