grandes-ecoles 2020 Q5

grandes-ecoles · France · x-ens-maths1__mp Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix

For $n \in \mathbb { N } ^ { \star }$, 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 $n \in \mathbb { N } ^ { \star }$, 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

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