= Symplectic adjoint
{title2=$\omega(Tv,w)=\omega(v,T^*w)$}
For a finite-dimensional <symplectic vector space>, the symplectic adjoint of a <linear map> $T$ is the unique <linear map> $T^*$ satisfying the displayed identity. Existence and uniqueness follow because the <nondegenerate bilinear form> identifies the <vector space> with its <dual space>. If $\omega$ has <matrix> $J$ in a <basis>, then $T^*=J^{-1}T^{\mathsf T}J$. Thus $(ST)^*=T^*S^*$ and $(T^*)^*=T$, and membership in the <symplectic Lie algebra> is equivalent to $T^*=-T$. This operation is linear over the ground <field>; it does not involve complex conjugation. It preserves <diagonalizability> and the property of being a <nilpotent linear map>, since it is conjugate to <matrix transpose>.
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