grandes-ecoles 2023 Q16

grandes-ecoles · France · mines-ponts-maths2__pc Proof Deduction or Consequence from Prior Results

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 $u : X \mapsto (I_N - K)X$, and for $X \in E$, the functions $\psi_X : t \mapsto H_t X$ and $\varphi_X : t \mapsto \|H_t X\|^2$. Deduce that $\varphi_X$ is differentiable and express $\varphi_X'(t)$ in terms of $q_u$.

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 $u : X \mapsto (I_N - K)X$, and for $X \in E$, the functions $\psi_X : t \mapsto H_t X$ and $\varphi_X : t \mapsto \|H_t X\|^2$.\\
Deduce that $\varphi_X$ is differentiable and express $\varphi_X'(t)$ in terms of $q_u$.

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