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I know that the property $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for inner product holds.

I have a matrix $A$, such that,

$\langle Ax, x \rangle = -\langle x, Ax \rangle$ i.e.

$\langle x, Ax \rangle = -\overline{\langle x, Ax \rangle}$.

What does this imply about $Ax$ and $x$?

I need to show that $\langle Ax, x \rangle = 0$, which is actually valid for any $A = -A^T$, which I verified using sample matrices, but cannot verify in general. Can anyone help?

Thanks.

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