Let $A$ be the matrix
$$A= \begin{pmatrix} 2 & -1 & 0\\ -1 & 3 & -1 \\ 0& -1 & 2 \end{pmatrix}$$ and $B$ a given vector in $\mathbb{R}^3$. Denote $U=(u_1,u_2,u_3)^t$ the solution of $AU=B$ and
$$U_1= \begin{pmatrix} u_1\\ 0\\ 0\end{pmatrix}, \ U_2= \begin{pmatrix} 0\\ u_2\\ 0\end{pmatrix}, \ U_3= \begin{pmatrix} 0\\ 0\\ u_3\end{pmatrix}.$$
Also denote $F(X)= \frac{1}{2} AX \cdot X - B \cdot X$ the corresponding functional of the linear system which is minimized by $U$.
I would like to prove the following inequality :
$$F(U) \geq F(U_1) + F(U_3).$$
I computed the difference, using the decomposition $U=U_1 + U_2 + U_3$ and the fact that $A$ is symmetric :
\begin{align*} F(U) - F(U_1) - F(U_3) & = A (U_1+U_3) \cdot U_2 - B \cdot U_2\\ &= A(U-U_2) \cdot U_2 - B \cdot U_2\\ &= A A^{-1} B \cdot U_2 - AU_2 \cdot U_2- B \cdot U_2 \text{ using the linear system}\\ &= -A U_2 \cdot U_2 \leq 0 \end{align*} since the matrix $A$ is symmetric and defined positively.
Finaly, I proved the exact inverse of the inequality I was looking for... Did I made a mistake somewhere ? Furthermore, I only used that $A$ was in $S_3^{++}(\mathbb{R})$, so I guess this result is true for any matrix in $S_3^{++}(\mathbb{R})$ (Update : I did use the values in the matrix to get the first equality)? Could it be generalized to $n \times n$ dimension matrices ? Is there any interpretation to this result ?