Editing Linear complex structure (section)

==Compatibility with other structures==

If {{math|''B''}} is a [[bilinear form]] on {{math|''V''}} then we say that {{math|''J''}} '''preserves''' {{math|''B''}} if
<math display="block">B(Ju, Jv) = B(u, v)</math>
for all {{math|''u'', ''v'' ∈ ''V''}}. An equivalent characterization is that {{math|''J''}} is [[skew-adjoint]] with respect to {{math|''B''}}:
<math display="block"> B(Ju,v) = -B(u,Jv). </math>

If {{math|''g''}} is an [[inner product]] on {{math|''V''}} then {{math|''J''}} preserves {{math|''g''}} if and only if {{math|''J''}} is an [[orthogonal transformation]]. Likewise, {{math|''J''}} preserves a [[nondegenerate]], [[skew-symmetric matrix|skew-symmetric]] form {{math|''ω''}} if and only if {{math|''J''}} is a [[symplectic transformation]] (that is, if <math display=inline> \omega(Ju,Jv) = \omega(u,v) </math>). For symplectic forms {{math|''ω''}} an interesting compatibility condition between {{math|''J''}} and {{math|''ω''}} is that
<math display=block> \omega(u, Ju) > 0 </math>
holds for all non-zero {{math|''u''}} in {{math|''V''}}. If this condition is satisfied, then we say that {{math|''J''}} '''tames''' {{math|''ω''}} (synonymously: that {{math|''ω''}} is '''tame''' with respect to {{math|''J''}}; that {{math|''J''}} is '''tame''' with respect to {{math|''ω''}}; or that the pair <math display="inline">(\omega,J)</math> is tame).

Given a symplectic form {{math|ω}} and a linear complex structure {{math|''J''}} on {{math|''V''}}, one may define an associated bilinear form {{math|''g''<sub>''J''</sub>}} on {{math|''V''}} by
<math display=block> g_J(u, v) = \omega(u, Jv). </math>
Because a [[symplectic form]] is nondegenerate, so is the associated bilinear form. The associated form is preserved by {{math|''J''}} if and only if the symplectic form is. Moreover, if the symplectic form is preserved by {{math|''J''}}, then the associated form is symmetric. If in addition {{math|''ω''}} is tamed by {{math|''J''}}, then the associated form is [[definite bilinear form|positive definite]]. Thus in this case {{math|''V''}} is an [[inner product space]] with respect to {{math|''g''<sub>''J''</sub>}}.

If the symplectic form {{math|ω}} is preserved (but not necessarily tamed) by {{math|''J''}}, then {{math|''g''<sub>''J''</sub>}} is the [[complex number|real part]] of the [[Hermitian form]] (by convention antilinear in the first argument) <math display=inline>h_J\colon V_J\times V_J\to\mathbb{C}</math> defined by
<math display=block> h_J(u,v) = g_J(u,v) + ig_J(Ju,v) = \omega(u,Jv) +i\omega(u,v). </math>