# Orthogonal projection

In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (kd)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. In particular, an orthogonal projection of a three-dimensional object onto a plane is obtained by intersections of planes drawn through all points of the object orthogonally to the plane of projection.

If such a projection leaves the origin fixed, it is a self-adjoint idempotent linear transformation; its matrix is a symmetric idempotent matrix. Conversely, every symmetric idempotent matrix is the matrix of the orthogonal projection onto its own column space. If M is a k×d matrix with d < k, the d columns spanning a d-dimensional subspace, then the matrix of the orthogonal projection onto the column space of M is

${\displaystyle P=M(M^{\top }M)^{-1}M^{\top }}$

(and before leaping to the conclusion that this must be an identity matrix, remember that M is not a square matrix but has more rows than columns!).

If the basis is orthonormal, the projection can be simplified to

${\displaystyle P=MM^{\top }}$

In functional analysis, the geometric notion is generalized as follows. An orthogonal projection is a bounded operator on a Hilbert space H which is self-adjoint and idempotent. It maps each vector v in H to the closest point of PH to v. PH is the range of P and it is a closed subspace of H.