orthogonal projection
Given the splitting V = W (+) W-perp, the orthogonal projection P onto W sends each vector to its W-component, dropping the part orthogonal to W. Geometrically, P v is the shadow of v on the subspace W, cast by light coming straight down perpendicular to W.
Two algebraic properties characterize it completely: P is idempotent (P^2 = P — projecting twice is the same as once) and self-adjoint (P^* = P). The self-adjointness is exactly what makes the projection orthogonal rather than oblique; an idempotent that is not self-adjoint projects along a slanted, non-perpendicular direction.
Its defining virtue is best approximation: among all vectors in W, P v is the unique closest one to v. The error v - P v lies in W-perp and is shorter than v - w for any other w in W (by the Pythagorean theorem). This is the engine behind least squares — the least-squares solution is just the orthogonal projection of the data onto the column space.
If q_1,...,q_k is an orthonormal basis of W, then P v = sum <v, q_j> q_j: add up the components along each basis direction. In matrix form with those columns in Q, P = Q Q^*. The eigenvalues of any orthogonal projection are only 0 and 1 — fully kept or fully discarded, nothing in between.
Idempotent and self-adjoint defines orthogonal projection; with an orthonormal basis of W it is the sum of components, or Q Q^* in matrix form.
Orthogonal vs oblique: P^2 = P alone is just a projection (could be slanted). Add P^* = P and the projection direction becomes perpendicular to its image — that is what orthogonal means here.