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 $c > 0$, where $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$.
Show that, for all $X \in \mathbb{R}^n$,
$${}^t X \Psi(M_c) X = {}^t X \Psi(M) X + 2c\, {}^t X \Psi(D) X + \frac{c^2}{2}\, {}^t X P X$$