nLab Hermitian form

Definition

Definition

(Hermitian form and Hermitian space)

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

• a complex-valued real-bilinear form

  $$h \;\colon\; V \otimes V \longrightarrow \mathbb{C}$$

such that this is symmetric sesquilinear, in that:

1. $h$ is complex-linear in the first argument;

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

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

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

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

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

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

Remark

A positive-definite and complete Hermitian vector space is called a Hilbert space.

Properties

General properties

Proposition

(basic properties of Hermitian forms)

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

1. the real part of the Hermitian form

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

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

   $$g \;\colon\; V \otimes V \to \mathbb{R}$$

2. the imaginary part of the Hermitian form

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

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

   $$\omega \;\colon\; V \wedge V \to \mathbb{R} \,.$$

hence

$$h = g - i \omega \,.$$

The two components are related by

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

Finally

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

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

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

These are called the Hermitian metrics.

Proof

The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

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

That $h$ is invariant under $J$ follows from its sesquilinarity

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

and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

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

and the relation between the two components:

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

as well as

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

Relation to Kähler spaces

Proposition

(relation between Kähler vector spaces and Hermitian spaces)

Given a real vector space $V$ with a linear complex structure $J$, then the following are equivalent:

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

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

where $\omega$ and $g$ are related by (1)

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

As $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-modules

While a non-degenerate inner product $(-\vert-)$ on a finite-dimensional real vector space $V$ is equivalently a linear isomorphism to its dual vector space

$$\array{ V &\overset{\sim}{\longrightarrow}& V^\ast &\overset{\sim}{\longrightarrow}& V \\ v &\mapsto& (v\vert-) &\mapsto& v }$$

the analogous statement for Hermitian complex inner products $\langle - \vert - \rangle$ fails, since the corresponding maps

$$\array{ \mathscr{H} &\overset{\sim}{\longrightarrow}& \mathscr{H} &\overset{\sim}{\longrightarrow}& \mathscr{H} \\ \vert \psi \rangle &\mapsto& \langle \psi \vert &\mapsto& \vert \psi \rangle }$$

are now complex anti-linear and hence not morphisms in the category of complex vector spaces.

What one does get is a complex-linear isomorphism to the anti-dual space?.

Another way to regard this situation is to observe that complex anti-linear involutions $\mathscr{H} \leftrightarrow \mathscr{H}^\ast$ on non-degenerate Hermitian spaces $\mathscr{H}$ are equivalently $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-module structures on the direct sum $\mathscr{H} \oplus \mathscr{H}^\ast$, regarded in the topos of $\mathbb{Z}/2$-sets, for $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$ the ring object given by the complex numbers equipped with their involution by complex conjugation:

Notice that these $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-modules arising from (non-degenerate, finite-dimensional) Hermitian vector spaces this way happen to carry also a complex structure, hence a compatible module-structure by the actual complex numbers (i.e. equipped with the trivial involution), given by $\underline{\mathrm{i}}$.

Using this, one may identify:

1. the space of linear operators on a Hermitian vector space as the equalizer of this imaginary rotation on the tensor square of the $(\mathbb{Z}/2 \curvearrowright \mathbb{C})$-module with the braiding and the identity,

2. among these that of hermitian operators as the further fixed locus of the involution action:

Related concepts

• Hilbert space

• bilinear form, quadratic form, sesquilinear form

• symplectic form, Kähler form

• hermitian matrix

• Hermitian manifold

• quadratic form

References

Among original articles:

• C. T. C. Wall, On the axiomatic foundations of the theory of Hermitian forms, Proc. Camb. Phil. Soc. 67 (1970) 243-250 [doi:10.1017/S0305004100045515, pdf]

See also:

• Wikipedia, Hermitian form

and see the references at Hilbert space.