# Orthogonal projection

In geometry, an **orthogonal projection** of a *k*-dimensional object onto a *d*-dimensional hyperplane (*d* < *k*) is obtained by intersections of (*k* − *d*)- 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

(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

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*.

See also spectral theorem, orthogonal matrix.

## Technical drawing

A concrete instance is used in technical drawing, where **orthogonal projection**, more correctly called orthographic projection, is drawing of the views of an object projected onto orthogonal planes. Commonly known views of this type are *plan* (*plan view*), *side view* and *elevation*.