In certain circumstances it turns out to be useful to pass from a complex vector space to a ‘corresponding’ real vector space or vice versa. The former is called realification and is really nothing more than restricting scalar multiplication to reals in the original complex vector space. If \(V\) is a vector space over \(\CC\), then its realification, \(V_\RR\), is the set \(V\) with vector addition and scalar multiplication by reals inherited unchanged from \(V\) (and the complex multiplication ‘forgotten’). If \(\{e_1,\dots,e_n\}\) is a basis of \(V\), then consider the vectors \(\{e_1,\dots,e_n,ie_1,\dots,ie_n\}\). It’s not difficult to see that these vectors form a basis of \(V_\RR\) so \(\dim V_\RR=2\dim V\). If \(T:V\mapto W\) is a linear transformation of complex vector spaces then we may also view it as a linear transformation, \(T_\RR:V_\RR\mapto W_\RR\), of real vector spaces. If the matrix representation of \(T\) with with respect to bases \(\{e_i\}\) and \(\{f_i\}\) of \(V\) and \(W\) respectively, is \(\mathbf{T}=\mathbf{A}+i\mathbf{B}\), with \(\mathbf{A}\) and \(\mathbf{B}\) both real matrices, then with respect to the bases \(\{e_1,\dots,e_n,ie_1,\dots,ie_n\}\) and \(\{f_1,\dots,f_n,if_1,\dots,if_n\}\), of respectively \(V_\RR\) and \(W_\RR\), \(T_\RR\) has the matrix representation,
\begin{equation}
\begin{pmatrix}
\mathbf{A}&-\mathbf{B}\\
\mathbf{B}&\mathbf{A}
\end{pmatrix}.
\end{equation}

This process of realification produces a rather special kind of real vector space, since it comes equipped with a particular linear operator which we’ll denote \(\iota\in\mathcal{L}(V_\RR)\), given by \(\iota v=iv\) and such that \(\iota^2=-\id_{V_\RR}\). This is the canonical example of a complex structure, a linear operator \(\iota\in\mathcal{L}(V)\) on any real vector space \(V\) such that \(\iota^2=-\id_V\).

Given a real vector space \(V\) equipped with a a complex structure \(\iota\) we may form a complex vector space structure on the set \(V\) by inheriting vector addition from \(V\) and defining complex scalar multiplication by \((a+ib)v=av+b\iota v\). It is not difficult to see that if \(\iota\) is the canonical complex structure on a real vector space \(V_\RR\) which is the realification of a complex vector space \(V\) then we just recover in this way the original complex vector space \(V\).

How might we proceed if we start from a real vector space without a complex structure? In particular, we would like to understand, in a basis free sense, the fact that whilst a real \(m\times n\) matrix \(\mathbf{A}\) is of course a linear transformation from \(\RR^n\) to \(\RR^m\), it may also be regarded as linear transformation from \(\CC^n\) to \(\CC^m\).

For any real vector space \(V\) we consider the (exterior) direct sum space \(V\oplus V\). We can then define complex structure on \(V\oplus V\), \(\iota:V\oplus V\mapto V\oplus V\) by \(\iota(v,v’)=(-v’,v)\). Then \(\iota^2=-\id_{V\oplus V}\), is clearly an isomorphism, and allows us to write any \((v,v’)\) in the suggestive form
\begin{equation*}
(v,v’)=(v,0)+\iota(v’,0).
\end{equation*}
The complexification of \(V\), \(V_\CC\), is defined to be the set \(V\oplus V\) equipped with the obvious vector addition and with complex multiplication given by
\begin{equation*}
(a+ib)(v,v’)\equiv(a+b\iota)(v,v’)=(av-bv’,av’+bv).
\end{equation*}
That \(V_\CC\) so defined is indeed a complex vector space is then obvious, given the properties of \(\iota\). It should also be clear that if \(V\) is an \(n\) dimensional real vector space, with basis \(\{e_i\}\) say, then the \((e_i,0)\) span \(V_\CC\). To see that they are also linearly independent over \(\CC\), suppose
\begin{equation*}
(a^1+ib^1)(e_1,0)+\dots+(a^n+ib^n)(e_n,0)=0.
\end{equation*}
This is true if and only if, \((a^ie_i,b^ie_i)=0\), itself true if and only if \(a^i=b^i=0\) for all \(i\).

Now, suppose \(T\in\mathcal{L}(V,W)\), is a linear transformation of real vector spaces. Define \(T_\CC:V_\CC\mapto W_\CC\) by \(T_\CC(v,v’)=(Tv,Tv’)\). Then since \(T_\CC(\iota(v,v’))=\iota(Tv,Tv’)\) it is clear that \(T_\CC\in\mathcal{L}(V_\CC,W_\CC)\). This is the unique linear map such that the diagram,
\begin{equation*}
\begin{CD}
V @>T>> W\\
@VV V @VV V\\
V_\CC @>T_\CC>> W_\CC
\end{CD}
\end{equation*}
in which the vertical maps are the standard inclusions (e.g.\ \(v\mapsto(v,0)\) of \(V\) into \(V_\CC\)), commutes. Given bases of \(V\) and \(W\), in which the matrix representation of \(T\) is \(\mathbf{T}\), it is clear that the matrix representation of \(T_\CC\) with respect to the bases obtained by the natural inclusions is identical.

Note that on \(V_\CC\) there is a natural notion of complex conjugation. For any \(v=(x,y)\in V_\CC\), \(x,y\in V\), we define \(v^*=(x,-y)\). For any \(T\in\mathcal{L}(V,W)\) we have \(T_\CC{(x,y)}^*=(T_\CC(x,y))^*\), that is, the complexification \(T_\CC\) commutes with this conjugation. In fact it is not difficult to show that a linear transformation in \(\mathcal{L}(V_\CC,W_\CC)\) is a complexification of a linear transformation in \(\mathcal{L}(V,W)\) if and only if it commutes with complex conjugation. Finally, note that \(\mathcal{L}(V,W)_\CC\cong\mathcal{L}(V_\CC,W_\CC)\), with the identification, \((S,T)\mapsto S_\CC+iT_\CC\), for \(S,T\in\mathcal{L}(V,W)\) and that therefore complex conjugation is also well defined in \(\mathcal{L}(V_\CC,W_\CC)\).