Let be $A$ and $B$ two real matrices of $n \times n$. And $\left \langle , \right \rangle$ denotes the usual inner product in $\mathbb{R}^{n}.$
Prove that if $A$ and $B$ are symmetric then $\forall x \in \mathbb{R}^{n}$ it satisfies:
\begin{align*} \left \langle (A^{2}+B^{2})x,x \right \rangle\geq \left \langle (AB+BA)x,x \right \rangle \end{align*} Hint: Consider $\left \langle (A-B)^{2}x,x \right \rangle$
What I think I can do is to note that:
\begin{align*} \left \langle A^{2}+B^{2})x,x \right \rangle&=\left \langle A^2x,x \right \rangle + \left \langle B^2x,x \right \rangle\\\left \langle AB+BA)x,x \right \rangle&=\left \langle AB,x \right \rangle + \left \langle BA,x \right \rangle \end{align*}
And then try to prove in general that:
\begin{align*} \left \langle A^2x,x \right \rangle&\geq\left \langle ABx,x \right \rangle\\ \end{align*}
Neverthless, I don't know how to use the hint and the fact that the matrices are symmetric. Can you help me please? I would really appreciate it.
