  The space of finite games can be decomposed into three orthogonal subspaces
[5], which are the subspaces of pure potential games, nonstrategic games and
pure harmonic games. The orthogonal projections onto these subspaces are
represented as the Moore-Penrose inverses of the corresponding linear operators
(i.e., matrices) [5]. Although the representation is compact and nice, no
analytic method is given to calculate Moore- Penrose inverses of these linear
operators. Hence using their results, one cannot verify whether a finite game
belongs to one of these subspaces. In this paper, jumping over calculating
Moore-Penrose inverses of these linear operators directly, via using group
inverses, in the framework of the semitensor product of matrices, we give
explicit polynomial representation for these orthogonal projections and for
potential functions of potential games. Using our results, one not only can
determine whether a finite game belongs to one of these subspaces, but also can
find an arbitrary finite game belonging to one of them. Besides, we give formal
definitions for these types of games by using their payoff functions. Based on
these results, more properties of finite games are revealed.

Optimization and Control