We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1. We denote by $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $K$ the application from $\Omega_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to a matrix $A$ $$K(A) = \mathbf{e} \cdot \mathbf{a}^T + \mathbf{a} \cdot \mathbf{e}^T - 2A$$ where $\mathbf{a}$ is the column matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are the diagonal coefficients of $A$. Show that for every matrix $A$ of $\Omega_n$ we have $K(A) \in \Delta_n$.
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1. We denote by $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $K$ the application from $\Omega_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to a matrix $A$
$$K(A) = \mathbf{e} \cdot \mathbf{a}^T + \mathbf{a} \cdot \mathbf{e}^T - 2A$$
where $\mathbf{a}$ is the column matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are the diagonal coefficients of $A$.
Show that for every matrix $A$ of $\Omega_n$ we have $K(A) \in \Delta_n$.