How does a gauge covariant derivative in a non-abelian field theory act on various quantities which are not valued in the algebra, and why? In particular, how does it act on a scalar valued function $f(x)$ and a matrix-valued function $\chi(x)$ which is not valued in the [representation of the] algebra of the gauge symmetry group, and why?
My understanding is that the gauge covariant derivative acts as follows on $\psi$, which is in the fundamental representation and would be written as a column matrix function, and $\phi$, which is in the adjoint representation and would be written as an nxn matrix function (let "$Id$" be the identity matrix, and absorb the coupling constant):
1) $D_{\mu} \psi = \partial_{\mu} \psi + i A_{\mu} \psi$, where $A_{\mu}$ is hermitian matrix-valued.
2) $D_{\mu} \chi = \partial \phi + i [A_{\mu}, \phi]$
So then for a matrix-valued function $\chi$ which is not valued in the matrix representation of the algebra, and a scalar-valued function $f(x)$, I expect:
3) $D_{\mu} \chi = \partial_{\mu} \chi + i A_{\mu} \chi$, using a matrix product in the second term, and
4) $D_{\mu} f = \text{Id} \hspace{.1cm} \partial_{\mu} f$. However, I am wondering if it is instead $D_{\mu} f = \partial_{\mu} (f \hspace{.1cm} \text{Id}) + i A_{\mu} (f \hspace{.1cm} \text{Id})$.
