Characteristic modes are very useful is gaining valuable insight into the electromagnetic behavior of material objects, independent of excitation. The foundational work for conducting objects was laid by R. J. Garbacz in 1965 and was and further developed by R. F. Harrington in 1971. Significant progress has been made over the last decade in extending the characteristic mode theory to penetrable objects. The determination of proper operators for penetrable objects which will not result in spurious modes was only resolved recently.

For homogeneous objects, the two most important ingredients of characteristic mode analysis are (i) a near-field operator that relates the surface currents to near-fields and (ii) a radiation operator that relates the surface currents to power radiated into the exterior space. In this talk we consider the operators associated with (i) a dipole antenna and (ii) lossy penetrable cylinder and elaborate on the steps needed to obtain a numerical solution. By considering global basis functions, the operators associated with M\"uller integral equations are shown to recover the familiar cylindrical harmonics for a lossy circular cylinder. Other shapes such as a square cylinder and an elliptical cylinder are also considered to generate the eigenvalues and eigenvectors via point patching or Galerkin methodologies. The dominant modes existing on all these structures and the physical significance of the resulting eigenvalues obtained from solving the generalized eigenvalue problem are discussed.