This paper is devoted to the study of an energy minimizing basis first introduced by Chan, Smith and Wan in 2000 for algebraic multigrid methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then proven to be locally homonic functions on each coarse grid ``elememt''. Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebaric multigrid method, the energy minimizing basis may also be applied for numerical homogenization.