Matrix and tensor constructions from a generic SU(n) vector

Projection operators appropriate to the general multispinor representations of SU(n) are constructed in a systematic way from the components of a single generic SU(n) vector transforming as the adjoint representation. The techniques have been devised with the problems of writing explicit forms for finite SU(n) rotations and nonlinear chiral Lagrangians kept specifically in mind. In particular, the general second rank tensors constructed from a single vector are found, counted, and exhibited in a very tractable form.