grandes-ecoles 2023 Q9

grandes-ecoles · France · mines-ponts-maths2__pc Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix

Let $E$ be a Euclidean space of dimension $N$. We denote by $(|)$ the inner product and $\|\cdot\|$ the associated Euclidean norm. Let $u$ be a self-adjoint endomorphism of $E$. We define $q_u : E \rightarrow \mathbf{R}$ by $q_u : x \mapsto (u(x) \mid x)$ and we assume that for all $x \in E$, $q_u(x) \geq 0$. State the spectral theorem for the endomorphism $u$. What can be said about the eigenvalues of $u$?

Let $E$ be a Euclidean space of dimension $N$. We denote by $(|)$ the inner product and $\|\cdot\|$ the associated Euclidean norm. Let $u$ be a self-adjoint endomorphism of $E$. We define $q_u : E \rightarrow \mathbf{R}$ by $q_u : x \mapsto (u(x) \mid x)$ and we assume that for all $x \in E$, $q_u(x) \geq 0$.\\
State the spectral theorem for the endomorphism $u$. What can be said about the eigenvalues of $u$?

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