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 $u : X \mapsto (I_N - K)X$, the orthogonal projection $p : E \rightarrow E$ onto $\ker(u)$, and for $X \in E$, the function $\varphi_X : t \mapsto \|H_t X\|^2$. We set $Y = X - p(X)$. We denote by $\lambda$ the smallest nonzero eigenvalue of $u$. Show that for all real $t \in \mathbf{R}_+$, $\varphi_Y'(t) \leq -2\lambda \varphi_Y(t)$. Deduce that $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$.
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 $u : X \mapsto (I_N - K)X$, the orthogonal projection $p : E \rightarrow E$ onto $\ker(u)$, and for $X \in E$, the function $\varphi_X : t \mapsto \|H_t X\|^2$.\\
We set $Y = X - p(X)$. We denote by $\lambda$ the smallest nonzero eigenvalue of $u$.\\
Show that for all real $t \in \mathbf{R}_+$, $\varphi_Y'(t) \leq -2\lambda \varphi_Y(t)$.\\
Deduce that $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$.