We explore the spatial profile of the ensemble average of the energy density of eigenchannels of the transmission matrix within random diffusive media using computer simulations and nonperturbative diagrammatic technique. A symmetrical profile with a peak in the middle of the sample is found for the fully transmitting eigenchannel and is shown to be closely related to a position dependent diffusion coefficient of the open media. We show that the average spatial profile of each transmission eigenchannel when normalized by the profile of the completely transmitting eigenchannel depends only upon the value of transmission through the corresponding eigenchannel. A universal expression for the average spatial profile is given in terms of the auxiliary localization lengths determined from transmission eigenvalues and position dependent diffusion coefficient. These lengths were first introduced by Dorokhov to describe the scaling of transmission and conductance through disordered media. Though direct measurement of energy distribution within a scattering medium is generally difficult, we demonstrate in microwave measurements that the integrated energy density stored in the media of each eigenchannel can be determined from the measurements of spectra of the transmission matrix. The derivative of the composite phase of the eigenchannels with respect to the angular frequency yields the contribution to the density of states (DOS) from the individual transmission eigenchannels. This is proportional to integrated energy stored and the dwell time of the transmission eigenchannel. The DOS determined from the transmission eigenchannel is shown to be in good agreement with DOS obtained by analyzing the field spectra into quasi-normal modes of the open medium. These results provide a path towards controlling the energy deposition within a scattering medium.