while reading a proof for conservation of energy of a certain system involving skew symmetric matrices (AT = -A), I came across this mathematical peculiarity that I cannot explain. Within the proof is the following statement:
$$2A\vec{x}\cdot\vec{x} = (A + A^T)\vec{x}\cdot\vec{x}$$
I tried this out with a sample skew symmetric matrix A, and indeed it works. But by inspection 2A cannot always equal A + AT (unless A = 0)! So my question is, am I violating some kind of matrix or dot product rule by trying to equate 2A and A + AT, and if so, what rule?
Sorry if this question comes across as ill-posed, I do not have a particularly sophisticated math background and would benefit most from simple answers, however possible.