grandes-ecoles 2010 QIIG1

grandes-ecoles · France · centrale-maths2__pc Matrices Structured Matrix Characterization
Determine the matrices $M$ of $\mathcal { M } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$. What are the matrices $M$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$?
Recall that $X _ { 0 } = \left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$, $H _ { 0 } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)$, $Y _ { 0 } = \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right)$, $J _ { 0 } = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$. 
Determine the matrices $M$ of $\mathcal { M } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$. What are the matrices $M$ of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$ that commute with $X _ { 0 }$?

Recall that $X _ { 0 } = \left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$, $H _ { 0 } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)$, $Y _ { 0 } = \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right)$, $J _ { 0 } = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$.
Paper Questions
QIA QIB QIC1 QIC2 QID1 QID2 QIE1 QIE2 QIIA1 QIIA2 QIIB QIIC QIID1 QIID2 QIIE1 QIIE2 QIIE3 QIIF1 QIIF2 QIIF3 QIIG1 QIIG2 QIIG3 QIIG4 QIIG5 QIIG6 QIIIA1 QIIIA2 QIIIA3 QIIIB1 QIIIB2 QIIIC1 QIIIC2 QIIIC3 QIIIC4 QIIIC5 QIIIC6