grandes-ecoles 2023 Q1

grandes-ecoles · France · x-ens-maths-a__mp Groups Algebra and Subalgebra Proofs

a) Show that $\mathbb{H}$ is a sub-$\mathbb{R}$-algebra of $M_2(\mathbb{C})$ stable under $Z \mapsto Z^*$. b) Let $Z \in \mathbb{H}$. Calculate $ZZ^*$ and deduce that every non-zero element of $\mathbb{H}$ is invertible. c) Let $Z \in \mathbb{H}$. Show that $Z \in \mathbb{R}_{\mathbb{H}}$ if and only if $ZZ' = Z'Z$ for all $Z' \in \mathbb{H}$.

a) Show that $\mathbb{H}$ is a sub-$\mathbb{R}$-algebra of $M_2(\mathbb{C})$ stable under $Z \mapsto Z^*$.\\
b) Let $Z \in \mathbb{H}$. Calculate $ZZ^*$ and deduce that every non-zero element of $\mathbb{H}$ is invertible.\\
c) Let $Z \in \mathbb{H}$. Show that $Z \in \mathbb{R}_{\mathbb{H}}$ if and only if $ZZ' = Z'Z$ for all $Z' \in \mathbb{H}$.

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