Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Set $P = QT$ as defined in question II.G.4. Let $Y ^ { \prime } \in \mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $( X , H , Y ^ { \prime } )$ is an admissible triple. a) Deduce from question II.G.1 the matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X$. b) Compute the matrices $\Phi _ { X } \left( Y - Y ^ { \prime } \right)$ and $\Phi _ { H } \left( Y - Y ^ { \prime } \right)$. c) Deduce that we have $Y ^ { \prime } = Y$.
Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Set $P = QT$ as defined in question II.G.4. Let $Y ^ { \prime } \in \mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $( X , H , Y ^ { \prime } )$ is an admissible triple.
a) Deduce from question II.G.1 the matrices of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X$.
b) Compute the matrices $\Phi _ { X } \left( Y - Y ^ { \prime } \right)$ and $\Phi _ { H } \left( Y - Y ^ { \prime } \right)$.
c) Deduce that we have $Y ^ { \prime } = Y$.