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Complex conjugate of a vector space
Complex conjugate of a vector space
Complex conjugate of a vector space
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In mathematics, the complex conjugate of a complex vector space is a complex vector space that has the same elements and additive group structure as but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of satisfies where is the scalar multiplication of and is the scalar multiplication of The letter stands for a vector in is a complex number, and denotes the complex conjugate of [1]

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

Motivation

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If and are complex vector spaces, a function is antilinear if With the use of the conjugate vector space , an antilinear map can be regarded as an ordinary linear map of type The linearity is checked by noting: Conversely, any linear map defined on gives rise to an antilinear map on

This is the same underlying principle as in defining the opposite ring so that a right -module can be regarded as a left -module, or that of an opposite category so that a contravariant functor can be regarded as an ordinary functor of type

Complex conjugation functor

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A linear map gives rise to a corresponding linear map that has the same action as Note that preserves scalar multiplication because Thus, complex conjugation and define a functor from the category of complex vector spaces to itself.

If and are finite-dimensional and the map is described by the complex matrix with respect to the bases of and of then the map is described by the complex conjugate of with respect to the bases of and of

Structure of the conjugate

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

The double conjugate is identical to

Complex conjugate of a Hilbert space

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Given a Hilbert space (either finite or infinite dimensional), its complex conjugate is the same vector space as its continuous dual space There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on is an inner multiplication to some fixed vector, and vice versa.[citation needed]

Thus, the complex conjugate to a vector particularly in finite dimension case, may be denoted as (v-dagger, a row vector that is the conjugate transpose to a column vector ). In quantum mechanics, the conjugate to a ket vector  is denoted as – a bra vector (see bra–ket notation).

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Complex conjugate of a vector space

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from Grokipedia
In mathematics, the complex conjugate of a complex vector space VV, often denoted Vˉ\bar{V}, is another complex vector space that shares the same underlying additive abelian group as VV but features a modified scalar multiplication given by λ⋅v=λˉv\lambda \cdot v = \bar{\lambda} v for scalars λ∈C\lambda \in \mathbb{C} and vectors v∈Vv \in V, where λˉ\bar{\lambda} is the complex conjugate of λ\lambda. This structure ensures Vˉ\bar{V} is isomorphic to VV as complex vector spaces via an antilinear bijection, typically the componentwise complex conjugation map c:V→Vˉc: V \to \bar{V} satisfying c(λv)=λˉc(v)c(\lambda v) = \bar{\lambda} c(v) and c2=idc^2 = \mathrm{id}. The construction of Vˉ\bar{V} is functorial, meaning it extends to a covariant on the category of complex vector spaces, preserving direct sums and tensor products up to isomorphisms, such as V⊕W‾≅Vˉ⊕Wˉ\overline{V \oplus W} \cong \bar{V} \oplus \bar{W} and V⊗W‾≅Vˉ⊗Wˉ\overline{V \otimes W} \cong \bar{V} \otimes \bar{W}. Applying the twice yields Vˉ‾≅V\overline{\bar{V}} \cong V, reflecting the involutive nature of complex conjugation. In finite dimensions, dim⁡CVˉ=dim⁡CV\dim_{\mathbb{C}} \bar{V} = \dim_{\mathbb{C}} V, and for representations of groups or algebras over C\mathbb{C}, the conjugate space corresponds to the representation with conjugated coefficients, often used to study real forms or self-conjugate representations. This concept plays a key role in several areas of , including , where it helps distinguish irreducible representations and compute characters (noting that the character of Vˉ\bar{V} is the complex conjugate of that of VV), and in , particularly for Hilbert spaces, where Vˉ\bar{V} facilitates the definition of sesquilinear forms and adjoints via conjugate linearity. For example, in the standard Cn\mathbb{C}^n, the conjugation map sends (z1,…,zn)(z_1, \dots, z_n) to (zˉ1,…,zˉn)(\bar{z}_1, \dots, \bar{z}_n), with fixed points under conjugation forming the real subspace Rn\mathbb{R}^n. More generally, the fixed points of a conjugation on VV form a real whose recovers VV.

Fundamentals

Definition

Given a complex vector space VV over the field C\mathbb{C}, the complex conjugate vector space, denoted V‾\overline{V}, consists of the same underlying set of elements and the same addition operation as VV. The scalar multiplication on V‾\overline{V} is twisted by complex conjugation: for λ∈C\lambda \in \mathbb{C} and v∈Vv \in V, the product is defined as λ⋅v=λ‾v\lambda \cdot v = \overline{\lambda} v, where λ‾\overline{\lambda} is the complex conjugate of λ\lambda and the multiplication on the right uses the original scalar multiplication of VV. This construction ensures that V‾\overline{V} is also a complex vector space. The additive group axioms hold unchanged from VV. For scalar multiplication axioms, distributivity over vector addition follows since λ⋅(u+v)=λ‾(u+v)=λ‾u+λ‾v=λ⋅u+λ⋅v\lambda \cdot (u + v) = \overline{\lambda} (u + v) = \overline{\lambda} u + \overline{\lambda} v = \lambda \cdot u + \lambda \cdot v. Distributivity over scalar addition holds as (λ+μ)⋅v=λ+μ‾v=(λ‾+μ‾)v=λ‾v+μ‾v=λ⋅v+μ⋅v(\lambda + \mu) \cdot v = \overline{\lambda + \mu} v = (\overline{\lambda} + \overline{\mu}) v = \overline{\lambda} v + \overline{\mu} v = \lambda \cdot v + \mu \cdot v, using the additivity of complex conjugation. Associativity of scalars with vectors is verified by λ⋅(μ⋅v)=λ⋅(μ‾v)=λ‾(μ‾v)=λ‾μ‾v=λμ‾v=(λμ)⋅v\lambda \cdot (\mu \cdot v) = \lambda \cdot (\overline{\mu} v) = \overline{\lambda} (\overline{\mu} v) = \overline{\lambda} \overline{\mu} v = \overline{\lambda \mu} v = (\lambda \mu) \cdot v, relying on the anti-automorphism property of conjugation (λμ‾=λ‾μ‾\overline{\lambda \mu} = \overline{\lambda} \overline{\mu}). The multiplicative identity satisfies 1⋅v=1‾v=v1 \cdot v = \overline{1} v = v, and compatibility of scalar addition with vector scalar multiplication follows analogously. Elements of V‾\overline{V} are typically identified with those of VV using the same symbols, with the scalar multiplication understood to be the twisted version; occasionally, to distinguish, an element v∈Vv \in V is denoted v‾∈V‾\overline{v} \in \overline{V}. The underlying real vector space structure is preserved, as real scalars r∈Rr \in \mathbb{R} satisfy r‾=r\overline{r} = r, so r⋅v=rvr \cdot v = r v matches the original real multiplication in VV.

Basic Properties

The complex conjugate of a VV, denoted Vˉ\bar{V}, preserves the of VV. Specifically, if {v1,…,vn}\{v_1, \dots, v_n\} is a basis for the finite-dimensional space VV over C\mathbb{C}, then {v1ˉ,…,vnˉ}\{\bar{v_1}, \dots, \bar{v_n}\} forms a basis for Vˉ\bar{V}, as the relations of and spanning carry over directly from VV under the identification of addition and the twisted λ⋅vˉ=λ‾ vˉ\lambda \cdot \bar{v} = \overline{\lambda} \, \bar{v}. This holds generally for infinite-dimensional cases as well, where bases or Hamel bases behave analogously. A key characterization involves linear maps between complex vector spaces. Consider a map f:V→Wf: V \to W. The associated conjugate map fˉ:Vˉ→W\bar{f}: \bar{V} \to W is defined by fˉ(vˉ)=f(v)‾\bar{f}(\bar{v}) = \overline{f(v)} for all v∈Vv \in V. To verify linearity of fˉ\bar{f} assuming ff is C\mathbb{C}-linear, first note additivity: fˉ(vˉ+wˉ)=f(v+w)‾=f(v)+f(w)‾=f(v)‾+f(w)‾=fˉ(vˉ)+fˉ(wˉ).\bar{f}(\bar{v} + \bar{w}) = \overline{f(v + w)} = \overline{f(v) + f(w)} = \overline{f(v)} + \overline{f(w)} = \bar{f}(\bar{v}) + \bar{f}(\bar{w}). For scalar multiplication, fˉ(λ⋅vˉ)=fˉ(λ‾ vˉ)=f(λ‾ v)‾=λ‾ f(v)‾=λ f(v)‾=λ fˉ(vˉ),\bar{f}(\lambda \cdot \bar{v}) = \bar{f}(\overline{\lambda} \, \bar{v}) = \overline{f(\overline{\lambda} \, v)} = \overline{\overline{\lambda} \, f(v)} = \lambda \, \overline{f(v)} = \lambda \, \bar{f}(\bar{v}), since λ‾ z‾=λ zˉ\overline{\overline{\lambda} \, z} = \lambda \, \bar{z} for λ∈C\lambda \in \mathbb{C} and z∈Wz \in W. Thus, fˉ\bar{f} is linear if ff is. Conversely, if fˉ\bar{f} is linear, define f(v)=fˉ(vˉ)‾f(v) = \overline{\bar{f}(\bar{v})}; then ff satisfies the linearity axioms by a symmetric argument, establishing the equivalence. Antilinear maps provide a complementary perspective. An antilinear map g:V→Wg: V \to W satisfies g(λv)=λ‾ g(v)g(\lambda v) = \overline{\lambda} \, g(v) and additivity for all λ∈C\lambda \in \mathbb{C}, v∈Vv \in V. Such maps correspond bijectively to linear maps from Vˉ\bar{V} to WW. Given gg, define h:Vˉ→Wh: \bar{V} \to W by h(vˉ)=g(v)h(\bar{v}) = g(v). Then hh is additive, and for scalars, h(λ⋅vˉ)=h(λ‾ vˉ)=g(λ‾ v)=λ g(v)=λ h(vˉ),h(\lambda \cdot \bar{v}) = h(\overline{\lambda} \, \bar{v}) = g(\overline{\lambda} \, v) = \lambda \, g(v) = \lambda \, h(\bar{v}), since g(λ‾ v)=λ g(v)g(\overline{\lambda} \, v) = \lambda \, g(v) by antilinearity and λ‾‾=λ\overline{\overline{\lambda}} = \lambda. The inverse correspondence sends a linear h:Vˉ→Wh: \bar{V} \to W to the antilinear g(v)=h(vˉ)g(v) = h(\bar{v}), yielding a canonical isomorphism between the spaces of antilinear maps V→WV \to W and linear maps Vˉ→W\bar{V} \to W. In the finite-dimensional case, linear operators on Vˉ\bar{V} admit matrix representations via entrywise conjugation. Suppose T:V→VT: V \to V is a linear operator with matrix A=(aij)A = (a_{ij}) relative to a basis {e1,…,en}\{e_1, \dots, e_n\} of VV, so T(ek)=∑jajkejT(e_k) = \sum_j a_{jk} e_j. The induced operator Tˉ:Vˉ→Vˉ\bar{T}: \bar{V} \to \bar{V} satisfies Tˉ(vˉ)=T(v)‾\bar{T}(\bar{v}) = \overline{T(v)}, and relative to the basis {e1ˉ,…,enˉ}\{\bar{e_1}, \dots, \bar{e_n}\}, Tˉ(ekˉ)=T(ek)‾=∑jajkej‾=∑jaˉjk ejˉ.\bar{T}(\bar{e_k}) = \overline{T(e_k)} = \overline{\sum_j a_{jk} e_j} = \sum_j \bar{a}_{jk} \, \bar{e_j}. Thus, the matrix of Tˉ\bar{T} is Aˉ=(aˉij)\bar{A} = (\bar{a}_{ij}), the entrywise complex conjugate of AA.

Motivation and Construction

Historical and Conceptual Motivation

The concept of the vector space arises from the need to handle antilinear operations within the framework of linear algebra over the complex numbers, particularly in fields like physics and where such maps emerge naturally but disrupt the standard category of linear transformations. In , operations such as or act antilinearly on state vectors, meaning they conjugate scalars rather than multiply them directly, which complicates their integration into linear operator algebras. Similarly, in of groups over C\mathbb{C}, irreducible representations may admit invariant sesquilinear forms that distinguish between real, complex, or quaternionic types, necessitating a construction that "linearizes" these antilinear symmetries to preserve the linearity of the underlying category. Historically, the idea of conjugate spaces gained prominence in the early through work on group representations, where the Frobenius-Schur indicator provided a tool to classify irreducible complex representations based on the existence of invariant bilinear or sesquilinear forms. Introduced by Georg Frobenius and in their 1906 paper on real representations of s, the indicator ν2(χ)=1∣G∣∑g∈Gχ(g2)\nu_2(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^2) determines whether a representation is realizable over the reals (ν2=1\nu_2 = 1), requires quaternions (ν2=−1\nu_2 = -1), or is of complex type (ν2=0\nu_2 = 0), implicitly relying on conjugation to relate the representation to its . This development built on Frobenius's earlier from the 1890s and Schur's extensions, marking an early recognition of conjugate structures as essential for bridging complex and real representations in finite group theory. From a category-theoretic perspective, the conjugate construction (−)‾:VectC→VectC\overline{(-)}: \mathbf{Vect}_\mathbb{C} \to \mathbf{Vect}_\mathbb{C} serves as a covariant that embeds antilinear maps into the linear category, allowing antilinear morphisms f:V→Wf: V \to W to be viewed as linear maps V→W‾V \to \overline{W} by adjusting on the codomain. This functoriality ensures that composition and identities preserve , facilitating the study of symmetries in complex vector spaces without leaving the linear framework, and it underlies adjunctions like Frobenius reciprocity in induced representations. A illustrative example from highlights this utility: the ⟨ψ∣\langle \psi | is an antilinear functional on the ket space of states ∣ψ⟩|\psi \rangle, as ⟨ψ∣(c∣ϕ⟩)=c∗⟨ψ∣ϕ⟩\langle \psi | (c |\phi \rangle) = c^* \langle \psi | \phi \rangle for complex cc, but it becomes a linear functional when viewed on the conjugate , aligning Dirac's duality with the linear structure of operators. This perspective, formalized in the 1930s, underscores how conjugate spaces resolve the antilinearity inherent in quantum inner products.

Functorial Construction

The conjugation functor, denoted ⋅‾:\VectC→\VectC\overline{\cdot}: \Vect_{\mathbb{C}} \to \Vect_{\mathbb{C}}, assigns complex VV its complex conjugate space Vˉ\bar{V}, which has the same underlying additive as VV but with scalar multiplication defined by α⋅vˉ:=α‾⋅vˉ\alpha \cdot \bar{v} := \overline{\alpha} \cdot \bar{v} for α∈C\alpha \in \mathbb{C} and vˉ∈Vˉ\bar{v} \in \bar{V}, where the overline on the right denotes the formal of elements of VV. This construction ensures Vˉ\bar{V} is a complex , as the twisted scalar multiplication preserves over C\mathbb{C}. The functor acts on morphisms by sending each C\mathbb{C}-linear map T:V→WT: V \to W to the C\mathbb{C}-linear map [Tˉ](/page/Linearmap):Vˉ→Wˉ[\bar{T}](/page/Linear_map): \bar{V} \to \bar{W} defined by Tˉ(vˉ)=T(v)‾\bar{T}(\bar{v}) = \overline{T(v)} for all v∈Vv \in V, where the overline on the right applies componentwise to the image under TT. To verify linearity of Tˉ\bar{T}, consider Tˉ(α⋅vˉ)=Tˉ(α‾⋅vˉ)=T(α‾⋅v)‾=α‾⋅T(v)‾=α⋅T(v)‾=α⋅Tˉ(vˉ)\bar{T}(\alpha \cdot \bar{v}) = \bar{T}(\overline{\alpha} \cdot \bar{v}) = \overline{T(\overline{\alpha} \cdot v)} = \overline{\overline{\alpha} \cdot T(v)} = \alpha \cdot \overline{T(v)} = \alpha \cdot \bar{T}(\bar{v}), and additivity follows similarly from the properties of TT. Functoriality requires that ⋅‾\overline{\cdot} preserves identities and composition. The identity map \idV:V→V\id_V: V \to V maps to \idVˉ=\idVˉ\bar{\id_V} = \id_{\bar{V}}, since \idVˉ(vˉ)=\idV(v)‾=vˉ\bar{\id_V}(\bar{v}) = \overline{\id_V(v)} = \bar{v}. For composition, if S:U→VS: U \to V and T:V→WT: V \to W are C\mathbb{C}-linear, then T∘S‾(uˉ)=(T∘S)(u)‾=T(S(u))‾=Tˉ(S(u)‾)=Tˉ(Sˉ(uˉ))=(Tˉ∘Sˉ)(uˉ)\overline{T \circ S}(\bar{u}) = \overline{(T \circ S)(u)} = \overline{T(S(u))} = \bar{T}(\overline{S(u)}) = \bar{T}(\bar{S}(\bar{u})) = (\bar{T} \circ \bar{S})(\bar{u}) for all u∈Uu \in U, so T∘S‾=Tˉ∘Sˉ\overline{T \circ S} = \bar{T} \circ \bar{S}. Thus, ⋅‾\overline{\cdot} is a covariant functor on the category of complex vector spaces and linear maps. Although covariant, the functor interchanges with the opposite category in the sense that it reverses the direction of scalar actions in certain dual constructions, such as Hom-spaces. In the finite-dimensional case, suppose VV and WW have bases {ei}\{e_i\} and {fj}\{f_j\}, respectively, and TT has matrix A=(aji)A = (a_{ji}) with respect to these bases, so T(ei)=∑jajifjT(e_i) = \sum_j a_{ji} f_j. Then Tˉ\bar{T} has matrix Aˉ=(aˉji)\bar{A} = (\bar{a}_{ji}), the entrywise of AA, with respect to the induced bases {eˉi}\{\bar{e}_i\} and {fˉj}\{\bar{f}_j\}. This matrix action underscores the functor's preservation of linear structure under conjugation.

Structural Aspects

Isomorphisms and Non-Naturality

For any complex vector space VV, there exists a C\mathbb{C}-linear ϕ:V→Vˉ\phi: V \to \bar{V}. Such an isomorphism can be constructed using a conjugation on VV, which is an antilinear involution c:V→Vc: V \to V satisfying c(v+w)=c(v)+c(w)c(v + w) = c(v) + c(w), c(zv)=zˉc(v)c(z v) = \bar{z} c(v), and c2=idc^2 = \mathrm{id}. The map cc induces a C\mathbb{C}-linear between VV and Vˉ\bar{V} by adjusting the structure accordingly. However, this isomorphism is not unique, as complex vector spaces generally admit multiple distinct conjugations, each yielding a different isomorphism. For instance, the space M2(C)M_2(\mathbb{C}) of 2×22 \times 2 complex matrices admits at least two non-equivalent conjugations, leading to different corresponding isomorphisms to its conjugate. In finite dimensions, an explicit isomorphism can be induced by a choice of basis. Let {ei}\{e_i\} be a basis for VV; then {ei}\{e_i\} also serves as a basis for Vˉ\bar{V}, since the underlying real dimension is preserved and the complex structures differ only in the action of scalars. The map ϕ\phi defined by sending each basis vector eie_i to the corresponding basis vector in Vˉ\bar{V} (adjusting for the conjugated scalar action via the basis coordinates) extends linearly to an ϕ:V→Vˉ\phi: V \to \bar{V}. Different choices of basis for VV produce different isomorphisms, underscoring the dependence on arbitrary selections. The conjugation functor, which sends VV to Vˉ\bar{V} and linear maps f:V→Wf: V \to W to their underlying real-linear extensions fˉ:Vˉ→Wˉ\bar{f}: \bar{V} \to \bar{W}, admits no to the identity functor on the category of complex vector spaces. Suppose, for contradiction, that there exists a α\alpha with components αV:V→Vˉ\alpha_V: V \to \bar{V} that are C\mathbb{C}-linear isomorphisms. Consider the scalar multiplication maps mλ:V→Vm_\lambda: V \to V given by v↦λvv \mapsto \lambda v for λ∈C\lambda \in \mathbb{C}; their images under the conjugation functor are mˉλ:Vˉ→Vˉ\bar{m}_\lambda: \bar{V} \to \bar{V} given by v↦λˉvv \mapsto \bar{\lambda} v. Naturality requires the V→mλVαV↓αV↓Vˉ→mˉλVˉ\begin{CD} V @>m_\lambda>> V \\ @V\alpha_V VV @V\alpha_V VV \\ \bar{V} @>>\bar{m}_\lambda> \bar{V} \end{CD}
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