This paper presents new results for a theory of eigenvalue modification through feedback gain vector parameter variation. A basis for the gain vector space is developed using large-scale perturbations of the eigenvalues such that movement of the gain vector along a direction in this space affects the position of only one real eigenvalue, or of a complex conjugate pair of eigenvalues. It is shown that this basis can be also used to directly compute the incremental sensitivities of the eigenvalues to the individual gain vector elements. Numerical examples are included to illustrate these theoretical concepts.