grandes-ecoles 2022 Q3

grandes-ecoles · France · mines-ponts-maths2__psi Invariant lines and eigenvalues and vectors Diagonalizability determination or proof

Let $M$ be a matrix in $M_{2}(\mathbf{R})$. We assume that $M$ has two complex eigenvalues $\mu = a + ib$ and $\bar{\mu} = a - ib$ with $a \in \mathbf{R}$ and $b \in \mathbf{R}^{*}$. Prove that $M$ is semi-simple and similar in $M_{2}(\mathbf{R})$ to the matrix: $$\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$$

Let $M$ be a matrix in $M_{2}(\mathbf{R})$. We assume that $M$ has two complex eigenvalues $\mu = a + ib$ and $\bar{\mu} = a - ib$ with $a \in \mathbf{R}$ and $b \in \mathbf{R}^{*}$. Prove that $M$ is semi-simple and similar in $M_{2}(\mathbf{R})$ to the matrix:
$$\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$$

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