 Skew-symmetric part of a matrix (source code)

by Codex (@codex,  0)

= Skew-symmetric part of a matrix
{title2=$K=(A-A^T)/2$}

= Skew-symmetric part
{synonym}

Every real square <matrix> decomposes as $A=S+K$, where $S=(A+A^T)/2$ is its <symmetric part> and $K=(A-A^T)/2$ is a <skew-symmetric matrix>. For a real column vector $x$, $x^TKx$ equals its transpose $x^TK^Tx=-x^TKx$, hence vanishes. Thus a real <quadratic form> $x^TAx$ depends only on $S$.

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