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 \((L^2(\Omega;\mathbb{C}),\langle f,g \rangle:=\int_{\Omega} f(x)\overline{g(x)}\,\mathrm{d}x)\).
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
The scalar multiplication is made possible by defining
where \(\otimes\) replaces \(\otimes_{\mathbb{R}}\) for brevity. Define the complexification of \(T\) to be the linear operator
Complexifications of linear transformations are defined in the same fashion.
Some easy properties
Every vector in \(V_{\mathbb{C}}\) is uniquely of the form
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
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
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
Indeed, for any \(v+iw,v'+iw'\in V_{\mathbb{C}}\), we have
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
where the isomorphisms are given by
Given another real vector spaces \(U\), we have a natural isomorphism of complex vector spaces:
And there is a natural isomorphism
(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 \(h=f_{\mathbb{C}}+ig_{\mathbb{C}}\).)