<s>[INST] if $a^2+b^2+c^2+2\rho(ab+bc+ca)\ge0$ then $\rho\ge-1/2$ This may be trivial but I am not able to prove that if $a^2+b^2+c^2+2\rho(ab+bc+ca)\ge0$ for $a,b,c\in\mathbb{R}$ then $\rho\ge-1/2$. Can anybody help me please?
Thanks!
 [/INST] With $M = \left(
\begin{array}{ccc}
 0 & \frac{1}{2} & \frac{1}{2} \\
 \frac{1}{2} & 0 & \frac{1}{2} \\
 \frac{1}{2} & \frac{1}{2} & 0 \\
\end{array}
\right)$ and $v = (a,b,c)$ we have
$$
v^{\top}I_3 v + 2\rho v^{\top}M v \ge 0\Rightarrow v^{\top}\left(I_3+2\rho M\right)v \ge0
$$
so choosing $\rho$ such that $I_3+2\rho M \gt 0$ (definite positiveness)
with the conditions in the sub-determinants
$$
1-\rho^2 > 0\\
(\rho -1)^2 (2 \rho +1) > 0
$$
follows in $$\rho \ge -\frac 12$$
</s>