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Hermitian form

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Linear algebra

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Definition

Definition

(Hermitian form and Hermitian space)

Let VV be a real vector space equipped with a complex structure J:VVJ\colon V \to V. Then a Hermitian form on VV is

  • a complex-valued real-bilinear form

    h:VV h \;\colon\; V \otimes V \longrightarrow \mathbb{C}

such that this is symmetric sesquilinear, in that:

  1. hh is complex-linear in the first argument;

  2. h(w,v)=(h(v,w)) *h(w,v) = \left(h(v,w) \right)^\ast for all v,wVv,w \in V

where () *(-)^\ast denotes complex conjugation.

A Hermitian form is positive definite (often assumed by default) if for all vVv \in V

  1. h(v,v)0h(v,v) \geq 0

  2. h(v,v)=0AAAAv=0h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0.

A complex vector space (V,J)(V,J) equipped with a (positive definite) Hermitian form hh is called a (positive definite) Hermitian space.

Properties

Genera properties

Proposition

(basic properties of Hermitian forms)

Let ((V,J),h)((V,J),h) be a positive definite Hermitian space (def. ). Then

  1. the real part of the Hermitian form

    g(,)Re(h(,)) g(-,-) \;\coloneqq\; Re(h(-,-))

    is a Riemannian metric, hence a symmetric positive-definite real-bilinear form

    g:VV g \;\colon\; V \otimes V \to \mathbb{R}
  2. the imaginary part of the Hermitian form

    ω(,)Im(h(,)) \omega(-,-) \;\coloneqq\; -Im(h(-,-))

    is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form

    ω:VV. \omega \;\colon\; V \wedge V \to \mathbb{R} \,.

hence

h=giω. h = g - i \omega \,.

The two components are related by

(1)ω(v,w)=g(J(v),w)AAAAAg(v,w)=ω(v,J(v)). \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(v)) \,.

Finally

h(J(),J())=h(,) h(J(-),J(-)) = h(-,-)

and so the Riemannian metrics gg on VV appearing from (and fully determining) Hermitian forms hh via h=giωh = g - i \omega are precisely those for which

(2)g(J(),J())=g(,). g(J(-),J(-)) = g(-,-) \,.

These are called the Hermitian metrics.

Proof

The positive-definiteness of gg is immediate from that of hh. The symmetry of gg follows from the symmetric sesquilinearity of hh:

g(w,v) Re(h(w,v)) =Re(h(v,w) *) =Re(h(v,w)) =g(v,w). \begin{aligned} g(w,v) & \coloneqq Re(h(w,v)) \\ & = Re\left( h(v,w)^\ast\right) \\ & = Re(h(v,w)) \\ & = g(v,w) \,. \end{aligned}

That hh is invariant under JJ follows from its sesquilinarity

h(J(v),J(w)) =ih(v,J(w)) =i(h(J(w),v)) * =i(i)(h(w,v)) * =h(v,w) \begin{aligned} h(J(v),J(w)) &= i h(v,J(w)) \\ & = i (h(J(w),v))^\ast \\ & = i (-i) (h(w,v))^\ast \\ & = h(v,w) \end{aligned}

and this immediately implies the corresponding invariance of gg and ω\omega.

Analogously it follows that ω\omega is skew symmetric:

ω(w,v) Im(h(w,v)) =Im(h(v,w) *) =Im(h(v,w)) =ω(v,w), \begin{aligned} \omega(w,v) & \coloneqq -Im(h(w,v)) \\ & = -Im\left( h(v,w)^\ast\right) \\ & = Im(h(v,w)) \\ & = - \omega(v,w) \,, \end{aligned}

and the relation between the two components:

ω(v,w) =Im(h(v,w)) =Re(ih(v,w)) =Re(h(J(v),w)) =g(J(v),w) \begin{aligned} \omega(v,w) & = - Im(h(v,w)) \\ & = Re(i h(v,w)) \\ & = Re(h(J(v),w)) \\ & = g(J(v),w) \end{aligned}

as well as

g(v,w) =Re(h(v,w) =Im(ih(v,w)) =Im(h(J(v),w)) =Im(h(J 2(v),J(w))) =Im(h(v,J(w))) =ω(v,J(w)). \begin{aligned} g(v,w) & = Re(h(v,w) \\ & = Im(i h(v,w)) \\ & = Im(h(J(v),w)) \\ & = Im(h(J^2(v),J(w))) \\ & = - Im(h(v,J(w))) \\ & = \omega(v,J(w)) \,. \end{aligned}

Relation to Kähler spaces

Proposition

(relation between Kähler vector spaces and Hermitian spaces)

Given a real vector space VV with a linear complex structure JJ, then the following are equivalent:

  1. ω 2V *\omega \in \wedge^2 V^\ast is a linear Kähler structure (def. );

  2. gVVg \in V \otimes V \to \mathbb{R} is a Hermitian metric (2)

where ω\omega and gg are related by (1)

ω(v,w)=g(J(v),w)AAAAAg(v,w)=ω(v,J(v)). \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(v)) \,.

References

  • C. T. C. Wall, On the axiomatic foundations of the theory of Hermitian forms, Proc. Camb. Phil. Soc. (1970), 67, 243

  • Wikipedia, Hermitian form

Last revised on December 21, 2017 at 08:37:40. See the history of this page for a list of all contributions to it.