In conceptual density functional theory, reactivity indexes as the Fukui function, the global hardness/softness, and hardness/softness kernels are fundamental linear responses extensively studied to predict the nucleophilic and electrophilic propensities of atoms in molecules. We demonstrate that the hardness/softness kernels of an isolated system can be expanded in eigenmodes, solutions of a variational principle. These modes are divided into two groups: the polarization modes and the charging modes. The eigenvectors of the polarization modes are orthogonal to the Fukui function and can be interpreted as densities induced at a constant chemical potential. The charging modes of an isolated system are associated with virtual charge transfers weighted by the Fukui function and obey an exact nontrivial sum rule. The exact relation between these charging eigenmodes and those of the polarizability kernel is established. The physical interpretation of the modes is discussed. Applications of the present findings to the Thomas-Fermi and von Weizacker kinetic energy functionals are presented. For a confined free quantum gas, described by the von Weizacker kinetic energy functional, we succeed to derive an approximate analytical solution for the Fukui function and for hardness/softness and polarizability kernels. Finally, we indicate how numerical calculations of the hardness kernel of a molecule could be performed from the Kohn-Sham orbitals.