[Math] Prove that $x^TAx=0$ when $x\in \mathbb{R}^n$ and $A$ is antisymmetric

linear algebramatrices

$A$ is real antisymmetric matrix which satisfies

$$ A^T=-A $$

Prove that

$$ x^TAx=0, \quad x \in \mathbb{R}^n$$

Example of real antisymmetric would be:

$$ A=\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix} $$

But other than this i dont know where to begin with this.

Now if someone can provide some insight on this that would be much appreciated.



Best Answer

Hint: use this$x^TAx=(x^TAx)^T=x^TA^Tx=-x^TAx$.

$x^TAx=(x^TAx)^T$ since $x^TAx$ is a scalar.

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