I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition of the principal symbol. Specifically, In Lawson and Michelsohn's Spin Geometry page 113 it says:
"Racall that the principal symbol of a differential operator $D:\Gamma (E) \to \Gamma (E)$ is a map which associates to each point $x \in X $ and each cotangent vector $\xi \in T^*_x(X)$, a linear map $\sigma _{\xi}(D):E_x \to E_x$ defined as follows. If in local coordinates we have $$ D=\sum_{|\alpha|\leq m}A_{\alpha}(x)\frac{\partial ^{|\alpha|}}{\partial x^{\alpha}} \text{ and } \xi=\sum_k \xi_k dx_k$$ where m is the order of $D$, then $$\sigma_{\xi}(D) = i^m \sum_{|\alpha|= m} A_{\alpha}(x)\xi^{\alpha}."$$ After going to the some other chapter you find out that $E$ is a complex vector bundle over $X$, a riemannian manifold, with a local trivialization $E|_U\to U \times \mathbb{C}^q$ and $A_{\alpha}(x)$ is a $q\times q$-matrix of smooth complex-valued functions.
So the question I have is if one is working with real vector bundles how does one define the principal symbol. I mean, if now one has that $A_{\alpha}(x)$ is a $q\times q$-matrix of smooth real-valued functions, how do you define the linear map $\sigma _{\xi}(D):E_x \to E_x$, because just taking the "$i^m$" factor off from the definition seems quite arbitrary. Any clarification is highly apreciated!