Covariant Derivative: Different for Vectors, Spinors & Matrices?

[thehangedman]

The covariant derivative is different in form for different tensors, depending on their rank.

What about other mathematical entities? The electromagnetic field A is a vector, but it has complex values. Is the covariant derivative different for complex valued vectors? And what about spinors? Matrices?

 

[haushofer]

For spinors you have to use the spin connection to define the covariant derivative, because tensors don't transform under spinoral representations of the Lorentz group. A matrix is not a tensor or spinorial object in general, so the cov. derivative on a matrix is not defined until you specify the spinorial/tensorial character.