# Skew-Symmetric Matirx

24/03/2011

Let , A is a skew-symmetric matrix if and only if

Proof:

Necessity: if A is a skew-symmetric matrix, then . Hence . Note , hence .

Sufficiency: if , then . I think we need the lemma that if A is a symmetric matrix, then if and only if . Since is a symmetric matrix, from we know . Hence A is skew-symmetric.

QED

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

The below is a simpler proof.

Since ,

, so ;

, so .

Hence .

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