目录
Complexification
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.
Every vector in \(V_{\mathbb{C}}\) is uniquely of the form
\[v+iw:=v\otimes 1+w\otimes i\quad (v,w\in V) \]Let \(\beta\) is a basis of the real vector space \(V\). Then \(\beta\otimes_{\mathbb{R}} 1\) is a basis of the complex vector space \(V_{\mathbb{C}}\). Also, \(T_{\mathbb{C}}\) is invertible iff \(T\) is invertible.
If \(V\) is nonzero and finite-dimensional, then: (i) \([T]_{\beta}=[T_{\mathbb{C}}]_{\beta\otimes 1}\); (ii) the characteristic/minimal polynomials of \(T\) and \(T_{\mathbb{C}}\) are equal; (iii) there exists a \(T\)-invariant subspace of dimension \(1\) or \(2\).
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 \(\langle v+iw,v'+iw' \rangle:=[\langle v,v' \rangle+\langle w,w' \rangle]+i[\langle w,v' \rangle-\langle v,w' \rangle]\) is the unique inner product on \(V_{\mathbb{C}}\) that restricts to \(\langle \cdot,\cdot \rangle\) on \(V\).
Realification
Let \(W\) be a complex vector space. Let \(\gamma\) be any basis of \(W\), then \(i\gamma\) is also a basis of \(W\). 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 \([S_{\mathbb{R}}]_{\gamma\cup i\gamma}=\begin{pmatrix}\text{Re} [S]_{\gamma} & -\text{Im}[S]_{\gamma} \\ \text{Im}[S]_{\gamma} & \text{Re} [S]_{\gamma}\end{pmatrix}\).
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 \(\langle w_1,w_2 \rangle_{\mathbb{R}}:=\text{Re}\langle w_1,w_2\rangle\) is an inner product such that \(\langle w,iw\rangle_{\mathbb{R}}=0\) for all \(w\in W_{\mathbb{R}}\).
Further Remarks
Complexification is 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 \(\text{Vect}_{\mathbb{R}}\) (c.f., Proposition 2.6.3 in Weibel's Introduction to Homological Algerbra).
There is a natural isomorphism
\[(V^*)_{\mathbb{C}}=V^*\otimes \mathbb{C}\cong \text{Hom}_{\mathbb{R}}(V,\mathbb{C})\cong \text{Hom}_{\mathbb{C}}(V_{\mathbb{c}},\mathbb{C})=(V_{\mathbb{C}})^* \]where the isomorphisms are given by
\[\varphi_1\otimes 1+\varphi_2\otimes i\leftrightarrow\underbrace{\varphi_1+i\varphi_2}_{=:\varphi}\leftrightarrow \Big(v\otimes z\mapsto z\varphi(v)\Big) \]Given another real vector spaces \(U\), we have
\[(U\otimes V)_{\mathbb{C}}\cong U_{\mathbb{C}}\otimes V_{\mathbb{C}} \]And there is a natural isomorphism
\[\begin{align*} (\text{Hom}_{\mathbb{R}}(U,V))_{\mathbb{C}}&\cong \text{Hom}_{\mathbb{C}}(U_{\mathbb{C}},V_{\mathbb{C}})\\ f\otimes 1+g\otimes i &\mapsto f_{\mathbb{C}}+ig_{\mathbb{C}} \end{align*} \]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 \(h=f_{\mathbb{C}}+ig_{\mathbb{C}}\).
标签：mathbb,langle,text,rangle,otimes,Complexification,gamma From： https://www.cnblogs.com/chaliceseven/p/16861843.html