quadratic programming /symmetric matrix

Question

I have a quadratic program with $ F: \mathbb{R^n} \rightarrow \mathbb{R}, F(x)=x^TQx$ I want to find a symmetric matrix M for Q, such that $F(x)=x^TMx$ holds for all x. I can write Q as sum of symmetric matrices and antisymmetric matrices: $ Q = \frac{1}{2}(Q + Q^T) +\frac{1}{2} (Q-Q^T)$ Is this the right track?

Answer 1

Yes.

Moreover, the antisymmetric matrix does not contribute. After all, $x^T Q x$ is a scalar and therefore $x^T Q x = (x^T Q x)^T = x^T Q^T x$ for any $x$. Consequently $x^T(Q-Q^T)x=0$.

So we can write: $$M=\frac 12(Q+Q^T)$$