Projectors on Intersections of Subspaces

Let P_L denote the orthogonal projector on a subspace L. Two constructions of projectors on intersections of subspaces are given in finite- dimensional spaces. One construction uses the singular value decomposition of P_LP_M to give an explicit formula for P_L⋂M. The other construction uses the result that the intersection of m ≥ 2 subspaces, L₁ ⋂ L₂ ⋂ … ⋂L_m, is the null-space of the matrix Q := Σ^m_i=1 λ_i (I − P_Li), for any positive coefficients {λ_i}. The projector P_{L1 ⋂ L2 ⋂ … ⋂Lm} can then be given in terms of the Moore- Penrose inverse of Q, or as the limit, as t → ∞, of the exponential function exp{−Qt}.