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Skew-Symmetric Matirx


Let A\in R^n, A is a skew-symmetric matrix if and only if x^TAx=0, \forall x\in R^n


Necessity: if A is a skew-symmetric matrix, then A^T=-A. Hence (x^TAx)^T=x^TA^Tx=-x^TAx. Note (x^TAx)^T=x^TAx, hence x^TAx=-x^TAx=0.

Sufficiency: if x^TAx=0, \forall x\in R^n, then x^T(A+A^T)x=0. I think we need the lemma that if A is a symmetric matrix, then x^TAx=0, \forall x\in R^n if and only if A=0. Since A+A^T is a symmetric matrix, from x^T(A+A^T)x=0 we know A+A^T=0. Hence A is skew-symmetric.


EDIT: The proof of sufficiency above is not correct. See here for a correct proof.

The below is a simpler proof.

Since x^TAx=0, \forall x,

e_i^TAe_i=0, so A_{ii}=0;

(e_i+e_j)^TA(e_i+e_j)=0, so A_{ij}+A_{ji}=0.

Hence A=-A^T.

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