We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. The hyperplane $\mathcal{H}$ has normal vector $Z$ and equation $x_1 + \cdots + x_n = 0$. Show that if $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$, then $$\forall X \in \mathcal{H}, \quad {}^t X \Psi(M) X \geqslant \lambda_{\min}\, {}^t XX \quad \text{and} \quad {}^t X \Psi(D) X \geqslant \mu_{\min}\, {}^t XX$$
We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. The hyperplane $\mathcal{H}$ has normal vector $Z$ and equation $x_1 + \cdots + x_n = 0$.
Show that if $\lambda_{\min}$ and $\mu_{\min}$ denote the respective minimal eigenvalues of $\Psi(M)$ and $\Psi(D)$, then
$$\forall X \in \mathcal{H}, \quad {}^t X \Psi(M) X \geqslant \lambda_{\min}\, {}^t XX \quad \text{and} \quad {}^t X \Psi(D) X \geqslant \mu_{\min}\, {}^t XX$$