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I've tried by starting with setting $x^tAx = 0 = x^t(-A^t)x$ and checking it termwise, but I don't think this will show me anything.

Could you explain how to approach this problem please?

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  • $\begingroup$ Do you mean $x^TAx = 0$ for all $x$? $\endgroup$ – астон вілла олоф мэллбэрг Mar 6 at 15:28
  • $\begingroup$ $A^t=-A$ is equivalent with $A^t+A=0$. This implies $x^t(A^t+A)x=0$ for all $x$. But this term is equivalent with $x^tAx=-(x^tAx)^t$. This equation looks like $y=-y$ ($y$ a real number) from which you can say $y=0$, since $y^t=y$ in $\mathbb{R}$. $\endgroup$ – Fakemistake Mar 6 at 19:35
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Hint: To prove the skew-symmetry of $\ A\ $ you need the equation to be true for all $\ x\ $. So what can you conclude about $\ x^t\left(A+A^t\right)y\ $ from the equation $\ 0 = \left(x+y\right)^tA\left(x+y\right)\ = x^tAx +x^tAy + y^tAx + y^tAy\ $?

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