We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$
Let $n \geqslant 2$ and $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n} \in \mathcal{B}_{n}(\mathbb{R})$ a matrix different from the identity. We denote $i$ the smallest integer belonging to $\llbracket 1,n \rrbracket$ such that $m_{i,i} \neq 1$. Show that there exists a matrix $M^{\prime} = \left(m_{i,j}^{\prime}\right)_{1 \leqslant i,j \leqslant n} \in \mathcal{B}_{n}(\mathbb{R})$ such that $f\left(M^{\prime}\right) \leqslant f(M)$ and $m_{j,j}^{\prime} = 1$ for every $j \in \llbracket 1,i \rrbracket$.