grandes-ecoles 2020 Q5

grandes-ecoles · France · x-ens-maths-a__mp_cpge Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix

For $n \in \mathbb { N } ^ { * }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

For $n \in \mathbb { N } ^ { * }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18