Polynomial representation for orthogonal projections onto subspaces of finite games

The space of finite games can be decomposed into three orthogonal subspaces [2] 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) [2], but no closed forms for these orthogonal projections are given. In this paper, in the framework of the semitensor product of matrices, we give explicit polynomial representation (actually closed forms) for these orthogonal projections.