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 $c > 0$. We consider $\gamma$ a real eigenvalue of the matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and $\binom{X_1}{X_2}$ an associated eigenvector. a) Show that ${}^t X_2 \Psi(M_\gamma) X_2 = 0$ and that $X_2 \neq 0$. Conclude that $\gamma \leqslant c^*$. b) What conclusion do we draw from this on the calculation of the smallest additive constant $c^*$?
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 $c > 0$. We consider $\gamma$ a real eigenvalue of the matrix $\left(\begin{array}{cc}0 & 2\Psi(M) \\ -I_n & -4\Psi(D)\end{array}\right)$ and $\binom{X_1}{X_2}$ an associated eigenvector.
a) Show that ${}^t X_2 \Psi(M_\gamma) X_2 = 0$ and that $X_2 \neq 0$. Conclude that $\gamma \leqslant c^*$.
b) What conclusion do we draw from this on the calculation of the smallest additive constant $c^*$?