Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Fix a matrix $Q \in GL(2, \mathbb{K})$ satisfying $X = QX_0Q^{-1}$. We define the vectors $u = Q \binom{1}{0}$ and $v = Q \binom{0}{1}$. a) By computing the vector $[ H , X ] u$ in two different ways, prove that $u$ is an eigenvector of the matrix $H$. b) By computing the vector $[ H , X ] v$ in two different ways, prove the existence of a scalar $t$ satisfying the identity: $H = Q \left( \begin{array} { c c } 1 & t \\ 0 & - 1 \end{array} \right) Q ^ { - 1 }$. c) Find a matrix $T \in GL ( 2 , \mathbb { K } )$ commuting with $X _ { 0 }$ and satisfying the relation $H = Q T H _ { 0 } ( Q T ) ^ { - 1 }$.
Let $X, H, Y$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ such that $(X, H, Y)$ forms an admissible triple. Fix a matrix $Q \in GL(2, \mathbb{K})$ satisfying $X = QX_0Q^{-1}$. We define the vectors $u = Q \binom{1}{0}$ and $v = Q \binom{0}{1}$.
a) By computing the vector $[ H , X ] u$ in two different ways, prove that $u$ is an eigenvector of the matrix $H$.
b) By computing the vector $[ H , X ] v$ in two different ways, prove the existence of a scalar $t$ satisfying the identity: $H = Q \left( \begin{array} { c c } 1 & t \\ 0 & - 1 \end{array} \right) Q ^ { - 1 }$.
c) Find a matrix $T \in GL ( 2 , \mathbb { K } )$ commuting with $X _ { 0 }$ and satisfying the relation $H = Q T H _ { 0 } ( Q T ) ^ { - 1 }$.