Complexification Keith Conrad

We want to describe a procedure for enlarging real vector spaces to complex vector spaces in a natural way. For instance, the natural complex analogues of Rn, Mn(R), and R[X] are Cn, Mn(C) and C[X]. Why do we want to complexify real vector spaces? One reason is related to solving equations. If we want to prove theorems about real solutions to a system of real linear equations or a system of real linear differential equations, it can be convenient as a first step to examine the complex solution space. Then we would try to use our knowledge of the complex solution space (for instance, its dimension) to get information about the real solution space. In the other direction, we may want to know if a subspace of Cn which is given to us as the solution set to a system of complex linear equations is also the solution set to a system of real linear equations. We will find a nice way to describe such subspaces once we understand the different ways that a complex vector space can occur as the complexification of a real subspace. We will give two descriptions of the complexification process, first in terms of a twofold direct sum (Section 2) and then in terms of tensor products (Section 3). The tensor product viewpoint is the more far-reaching one, but seeing how the direct sum method of complexification expresses everything may help convince the reader that there is nothing unexpected about this use tensor products. Moreover, idiosyncratic aspects of the direct sum construction will turn out to be completely natural features of the tensor product construction. After comparing the two constructions, we will see (Section 4) how to use a special structure on a complex vector space, called a conjugation, to describe the real subspaces which have the given complex vector space as its complexification. References on this topic are [1, pp. 79–81], [2, §77], and [3, pp. 262–263]. In [1] and [3] tensor products are used, while [2] uses direct sums.