Symmetric part of A contributes to quadratic form

Question

In my statistics note, when it talks about quadratic forms, it goes on saying:

"$x^tAx=\frac12x^t(A+A^t)x$ implies that only the symmetric part of A contributes to the quadratic form."

I am having hard time understanding the meaning of the sentence. I understand $(A + A^t)$ is symmetric but what it means when it says "only symmetric part contributes to quadratic form"?

Answer 1

In other words: take any matrix $A$, and any anti-symmetric matrix $B$. For all real vectors $x$, we have $$ x^TA x = x^T(A + B)x $$ That is, changing the anti-symmetric part of a matrix does not change the resulting quadratic form. So, the quadratic form is determined entirely from the symmetric part.