Complexification

Complexification

The complexification of the real vector space \(\mathbb{R}^n\) is the complex vector space \(\mathbb{C}^n\). The complexification of the real inner product space \((L^2(\Omega;\mathbb{R}),\langle u,v \rangle:=\int_{\Omega} u(x)v(x)\,\mathrm{d}x)\) is the complex inner product space \((L^2(\Omega;\mathbb{C}),\langle f,g \rangle:=\int_{\Omega} f(x)\overline{g(x)}\,\mathrm{d}x)\).

目录

• Complexification
  • What does "complexification" mean ?
  • Some easy properties
  • Inner product attached
  • The concept of realification
  • Functorial aspect

What does "complexification" mean ?

Let \(V\) be a real vector space and \(T\) a linear operator on \(V\). Define the complexification of \(V\) to be the complex vector space

\[V_{\mathbb{C}}:=V\otimes_{\mathbb{R}}\mathbb{C} \]

The scalar multiplication is made possible by defining

\[\lambda (v\otimes \mu):=v\otimes (\lambda\mu)\quad (v\in V;\lambda,\mu\in \mathbb{C}) \]

where \(\otimes\) replaces \(\otimes_{\mathbb{R}}\) for brevity. Define the complexification of \(T\) to be the linear operator

\[T_{\mathbb{C}}:=T\otimes \text{id}_{\mathbb{C}} \]

Complexifications of linear transformations are defined in the same fashion.

Some easy properties

Every vector in \(V_{\mathbb{C}}\) is uniquely of the form

\[v+iw:=v\otimes 1+i(w\otimes 1)=v\otimes 1+w\otimes i\quad (v,w\in V) \]

If \(\beta\) is a basis for the real vector space \(V\), then \(\beta\otimes_{\mathbb{R}} 1\) is automatically a basis for the complex vector space \(V_{\mathbb{C}}\). In particular, \(\dim_{\mathbb{C}}(V_{\mathbb{C}})=\dim_{\mathbb{R}}(V)\). It is obvious that if \(T\) is invertible, then so is \(T_{\mathbb{C}}\), and

\[(T_{\mathbb{C}})^{-1}=(T^{-1})_{\mathbb{C}} \]

Moreover, if \(V\) is nonzero and finite-dimensional, then i) thanks to the fact that \(T_{\mathbb{C}}\) has an eigenvector, there exists a \(T\)-invariant subspace of dimension \(1\) or \(2\); ii) \([T_{\mathbb{C}}]_{\beta\otimes 1}=[T]_{\beta}\), and hence the characteristic polynomials of \(T_{\mathbb{C}}\) and \(T\) are equal.

Inner product attached

If \(\langle\cdot,\cdot\rangle:V\times V\to \mathbb{R}\) is an inner product, then \(\langle\cdot,\cdot\rangle_{\mathbb{C}}:V_{\mathbb{C}}\times V_{\mathbb{C}}\to \mathbb{C}\) defined by

\[\langle v+iw,v'+iw' \rangle_{\mathbb{C}}:=\langle v,v' \rangle+\langle w,w' \rangle+i\langle w,v' \rangle-i\langle v,w' \rangle \]

is the unique inner product on \(V_{\mathbb{C}}\) that restricts back to \(\langle \cdot,\cdot \rangle\). We claim that, with respect to this pair of inner products, if \(T\) has an adjoint, then so does \(T_{\mathbb{C}}\), and

\[(T_{\mathbb{C}})^*=(T^*)_{\mathbb{C}} \]

Indeed, for any \(v+iw,v'+iw'\in V_{\mathbb{C}}\), we have

\[\begin{align*} &\hphantom{=\ }\langle T_{\mathbb{C}}(v+iw),v'+iw' \rangle_{\mathbb{C}}\\ &=\langle T(v)+iT(w),v'+iw' \rangle_{\mathbb{C}}\\ &=\langle T(v),v' \rangle+\langle T(w),w' \rangle+i\langle T(v),w' \rangle-i\langle T(w),v' \rangle\\ &=\langle v,T^*(v') \rangle+\langle w,T^*(w') \rangle+i\langle v,T^*(w') \rangle-i\langle w,T^*(v') \rangle\\ &=\langle v+iw,T^*(v')+iT^*(w') \rangle_{\mathbb{C}}\\ &=\langle v+iw,(T^*)_{\mathbb{C}}(v'+iw') \rangle_{\mathbb{C}} \end{align*} \]

The concept of realification

Let \(W\) be a complex vector space. Let \(\gamma\) be any basis for \(W\), then \(i\gamma\) is also a basis for \(W\). Clearly, \(\gamma\cap i\gamma=\varnothing\), and \(\gamma\cup i\gamma\) is linearly independent over \(\mathbb{R}\). Define \(W_{\mathbb{R}}\) to be the real vector space formed by all the linear combinations with real coefficients of the vectors in \(\gamma\cup i\gamma\). Since \(W_{\mathbb{R}}=\text{span}_{\mathbb{R}}(\gamma)\oplus \text{span}_{\mathbb{R}}(i\gamma)\), it is independent of the choice of \(\gamma\), and is called the realification of \(W\). They have the same underlying set. Let \(S\) be a linear operator on \(W\), then it is automatically a linear operator on \(W_{\mathbb{R}}\), denoted by \(S_{\mathbb{R}}\). If \(W\) is nonzero and finite-dimensional, then \(\dim_{\mathbb{R}}(W_{\mathbb{R}})=2\dim_{\mathbb{C}}(W)\), and \([S_{\mathbb{R}}]_{\gamma\cup i\gamma}=\begin{pmatrix}\Re [S]_{\gamma} & -\Im[S]_{\gamma} \\ \Im[S]_{\gamma} & \Re [S]_{\gamma}\end{pmatrix}\). If \(\langle \cdot,\cdot \rangle:W\times W\to \mathbb{C}\) is an inner product, then \(\langle \cdot,\cdot \rangle_{\mathbb{R}}:W_{\mathbb{R}}\times W_{\mathbb{R}}\to \mathbb{R}\) defined by \(\langle w_1,w_2 \rangle_{\mathbb{R}}:=\Re\langle w_1,w_2\rangle\) is the unique inner product such that \(\langle w,w \rangle_{\mathbb{R}}=\langle w,w \rangle\) and \(\langle w,iw\rangle_{\mathbb{R}}=0\) for all \(w\in W_{\mathbb{R}}\).

Functorial aspect

Complexification is obviously an additive functor from \(\text{Vect}_{\mathbb{R}}\) to \(\text{Vect}_{\mathbb{C}}\). By knowledge of homological algebra, it is the left adjoint functor of the forgetful functor from \(\text{Vect}_{\mathbb{C}}\) to \(\text{Vect}_{\mathbb{R}}\) (see, for example, Proposition 2.6.3 in Weibel's Introduction to Homological Algerbra). We now present some natural isomorphisms. The first is

\[(V^*)_{\mathbb{C}}=V^*\otimes \mathbb{C}\cong \text{Hom}_{\mathbb{R}}(V,\mathbb{C})\cong \text{Hom}_{\mathbb{C}}(V_{\mathbb{c}},\mathbb{C})=(V_{\mathbb{C}})^* \]

where the isomorphisms are given by

\[\varphi_1\otimes 1+\varphi_2\otimes i\longleftrightarrow\underbrace{\varphi_1+i\varphi_2}_{=:\varphi}\longleftrightarrow \Big(v\otimes 1\mapsto \varphi(v)\Big) \]

Given another real vector spaces \(U\), we have a natural isomorphism of complex vector spaces:

\[\begin{align*} U_{\mathbb{C}}\otimes_{\mathbb{C}} V_{\mathbb{C}} & \overset{\cong}{\longrightarrow} (U\otimes_{\mathbb{R}} V)_{\mathbb{C}} \end{align*} \]

And there is a natural isomorphism

\[\begin{align*} (\text{Hom}_{\mathbb{R}}(U,V))_{\mathbb{C}}&\overset{\cong}{\longrightarrow} \text{Hom}_{\mathbb{C}}(U_{\mathbb{C}},V_{\mathbb{C}})\\ f\otimes 1+g\otimes i &\mapsto f_{\mathbb{C}}+ig_{\mathbb{C}} \end{align*} \]

(For any \(h\in \text{Hom}_{\mathbb{C}}(U_{\mathbb{C}},V_{\mathbb{C}})\), there exists a unique pair of maps \(f,g:U\to V\) such that \(h(u\otimes 1)=f(u)\otimes 1+g(u)\otimes i\) for all \(u\in U\). It is easy to check that \(f,g\) are linear and \(h=f_{\mathbb{C}}+ig_{\mathbb{C}}\).)