We define a new set of indices for a generalized linear system. These indices, referred to as the homogeneous indices, are a natural generalization of the minimal column indices (Kronecker indices) of an ordinary state-space system. We prove that the homogeneous indices are a complete set of invariants for the action of a natural group of feedback transformations on generalized linear systems. We also show that the homogeneous indices determine exactly which closed loop invariant polynomials can be assigned by feedback, thereby generalizing the Control Structure Theorem of Rosenbrock.