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 and n×k is a matrix with more rows than columns, the k columns spanning a k-dimensional subspace of an n-dimensional space, then the matrix of the orthogonal projection onto the column space of M is

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.