grandes-ecoles 2019 Q8

grandes-ecoles · France · centrale-maths1__pc Matrices Diagonalizability and Similarity

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\Gamma(\mathbb{R})$ is not a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{R})$.

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.

Show that $\Gamma(\mathbb{R})$ is not a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{R})$.

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