= Symmetric part determines a real quadratic functional
{title2=$S=(L+L^*)/2$}
For a bounded operator on a real <Hilbert space>, $\langle Lv,v\rangle=\langle Sv,v\rangle$ with $S=(L+L^*)/2$. Consequently the <Euler-Lagrange equation> of $\langle Lv,v\rangle-2\langle f,v\rangle$ is $Sv=f$. It is $Lv=f$ only when $L$ is <self-adjoint> or when the relevant solution also annihilates the skew part. For example $L=\begin{pmatrix}1&-1\\1&1\end{pmatrix}$ has positive quadratic form $\|v\|^2$, but that form contains no information about its skew part.
Back to article page
