The linear and nonlinear optical properties of pure 3,3'-bis(3-carboxypropyl)-5,5',6,6'- tetrachloro-1,1'-dioctylbenzimidacarbocyanine (C8O3) aggregates as well as of C8O3 aggregates in the presence of PVA are studied at room temperature. The obtained results reveal that absorption spectra of both aggregates are governed by four absorption bands. Three of these represent transitions having dipole moments oriented parallel to the axis of the aggregate and one oriented perpendicular to it. The two lower energy parallel transitions are found to be weakly coupled which suggests that they represent two excitons located on different walls of the double-wall cylinder. In contrast, the lowest energy parallel transition and the transition having a transition dipole moment oriented perpendicular to the aggregate axis share the same ground state which reveals that they belong to the same exciton band. The manner of modification of the linear absorption with addition of PVA as well as the similarity of the pump-probe spectra of the two investigated aggregates suggests that this exciton band originates from the inner wall of the double-wall cylinder.