Complex conjugate of a vector space

Template:Short description In mathematics, the complex conjugate of a complex vector space $V$ is a complex vector space $\bar{V}$ that has the same elements and additive group structure as $V,$ but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of $\bar{V}$ satisfies $$\alpha *v=\bar{\alpha }\cdot v$$ where $*$ is the scalar multiplication of $\bar{V}$ and $\cdot$ is the scalar multiplication of $V.$ The letter $v$ stands for a vector in $V,$ $\alpha$ is a complex number, and $\bar{\alpha }$ denotes the complex conjugate of $\alpha .$

More concretely, the complex conjugate vector space is the same underlying Template:Em vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure $J$ (different multiplication by $i$).

Motivation

If $V$ and $W$ are complex vector spaces, a function $f:V\to W$ is antilinear if $$f(v+w)=f(v)+f(w)\text{ and }f(\alpha v)=\bar{\alpha }f(v)$$ With the use of the conjugate vector space $\bar{V}$, an antilinear map $f:V\to W$ can be regarded as an ordinary linear map of type $\bar{V}\to W.$ The linearity is checked by noting: $$f(\alpha *v)=f(\bar{\alpha }\cdot v)=\overline{\stackrel{‾}{\alpha }}\cdot f(v)=\alpha \cdot f(v)$$ Conversely, any linear map defined on $\bar{V}$ gives rise to an antilinear map on $V.$

This is the same underlying principle as in defining the opposite ring so that a right $R$-module can be regarded as a left $R^{op}$-module, or that of an opposite category so that a contravariant functor $C\to D$ can be regarded as an ordinary functor of type $C^{op}\to D.$

Complex conjugation functor

A linear map $f:V\to W$ gives rise to a corresponding linear map $\bar{f}:\bar{V}\to \bar{W}$ that has the same action as $f.$ Note that $\bar{f}$ preserves scalar multiplication because $$\bar{f}(\alpha *v)=f(\bar{\alpha }\cdot v)=\bar{\alpha }\cdot f(v)=\alpha *\bar{f}(v)$$ Thus, complex conjugation $V\mapsto \bar{V}$ and $f\mapsto \bar{f}$ define a functor from the category of complex vector spaces to itself.

If $V$ and $W$ are finite-dimensional and the map $f$ is described by the complex matrix $A$ with respect to the bases $ℬ$ of $V$ and $𝒞$ of $W,$ then the map $\bar{f}$ is described by the complex conjugate of $A$ with respect to the bases $\overline{ℬ}$ of $\bar{V}$ and $\overline{𝒞}$ of $\bar{W}.$

Structure of the conjugate

The vector spaces $V$ and $\bar{V}$ have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from $V$ to $\bar{V}.$

The double conjugate $\overline{\stackrel{‾}{V}}$ is identical to $V.$

Complex conjugate of a Hilbert space

Given a Hilbert space $\mathscr{H}$ (either finite or infinite dimensional), its complex conjugate $\bar{\mathscr{H}}$ is the same vector space as its continuous dual space ${\mathscr{H}}^{\prime }.$ There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on $\mathscr{H}$ is an inner multiplication to some fixed vector, and vice versa.Template:Citation needed

Thus, the complex conjugate to a vector $v,$ particularly in finite dimension case, may be denoted as $v^{†}$ (v-dagger, a row vector that is the conjugate transpose to a column vector $v$). In quantum mechanics, the conjugate to a ket vector $|\psi ⟩$ is denoted as $⟨\psi |$ – a bra vector (see bra–ket notation).

See also

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• conjugate bundle

Further reading

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. Template:ISBN. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).