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 $c > 0$. Let $c^* = \alpha^*$ be the minimal constant found previously, and $X^*$ the associated vector. a) Show that $\Psi(M_{c^*}) X^* = 0$. We set $Y^* = \frac{2}{c^*} \Psi(M) X^*$. b) Show that the column vector $\binom{Y^*}{X^*}$ is an eigenvector of the $2n \times 2n$ matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and that $c^*$ is an eigenvalue of this matrix.
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 $c > 0$. Let $c^* = \alpha^*$ be the minimal constant found previously, and $X^*$ the associated vector.
a) Show that $\Psi(M_{c^*}) X^* = 0$.
We set $Y^* = \frac{2}{c^*} \Psi(M) X^*$.
b) Show that the column vector $\binom{Y^*}{X^*}$ is an eigenvector of the $2n \times 2n$ matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and that $c^*$ is an eigenvalue of this matrix.