Change summation to matrix form

Question

Is it possible to change the following summation into matrix form, so that the summation sign disappears?

$\sum_{i,j=1}^{n}x_ix_j(\Omega x)_i(\Omega x)_j$

where $x \in \mathbb R^{n\times 1} $ and $\Omega \in \mathbb R^{n\times n}$.

Answer 1

Note that $x_j$ is just a real number, so it commutes with $(\Omega x)_i$ matrix. Also note that the $i$ and $j$ indexes are completely independent, so one can write $$\sum_{i=1}^nx_i(\Omega x)_i\sum_{j=1}^nx_j(\Omega x)_j=[x\cdot(\Omega x)]^2$$