grandes-ecoles 2023 Q17

grandes-ecoles · France · mines-ponts-maths2__pc Proof Proof of Stability or Invariance

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$ with $\ker(u) = \operatorname{Vect}(U)$, and the matrix $H_t$. We denote by $p : E \rightarrow E$ the orthogonal projection onto $\ker(u)$. Let $t \in \mathbf{R}_+$. Show that $p(H_t X) = p(X)$.

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$ with $\ker(u) = \operatorname{Vect}(U)$, and the matrix $H_t$.\\
We denote by $p : E \rightarrow E$ the orthogonal projection onto $\ker(u)$.\\
Let $t \in \mathbf{R}_+$. Show that $p(H_t X) = p(X)$.

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