= Quadratic-form criterion for a skew-symmetric matrix
{title2=$A^\top=-A\iff x^\top Ax=0\text{ for every real }x$}
\b[Proof.] For a <skew-symmetric matrix>, the scalar $x^\top Ax$ equals its <transpose> $x^\top A^\top x=-x^\top Ax$, so it vanishes. Conversely, substituting <standard basis> <vectors> $e_i$ into the vanishing <quadratic form> gives $a_{ii}=0$. Substituting $e_i+e_j$ gives $a_{ij}+a_{ji}=0$. Thus $A^\top=-A$. In particular, a real <quadratic form> depends only on a <matrix>'s <symmetric part>.
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